$sl{}\left(2\right)$-subalgebras of so(17), type $B_8$

Number of sl(2) subalgebras: 85.

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Length longest root ambient algebra squared/4= 1/2

Given a root subsystem $P$, and a root sub-subsystem ${P}_{0}$, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant $e{}\left(P, {P}_{0}\right)$ (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an $sl{}\left(2\right)$ is an $S$-subalgebra in the root subalgebra generated by $P$, such that it has characteristic 2 for all simple roots of $P$ lying in ${P}_{0}$, then $e{}\left(P, {P}_{0}\right)=0$.

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H that wasn't realized on the first attempt but was ultimately realized: (12, 22, 30, 36, 40, 42, 44, 44), Type: ${A^{184}}_{1}$.
It turns out that in the current case of Cartan element h = (12, 22, 30, 36, 40, 42, 44, 44) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (10, 18, 24, 28, 32, 34, 36, 36), Type: ${A^{120}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 18, 24, 28, 32, 34, 36, 36) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (10, 18, 24, 28, 30, 32, 34, 34), Type: ${A^{114}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 18, 24, 28, 30, 32, 34, 34) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (10, 18, 24, 28, 30, 32, 32, 32), Type: ${A^{112}}_{1}$.
It turns out that in the current case of Cartan element h = (10, 18, 24, 28, 30, 32, 32, 32) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (8, 14, 20, 24, 28, 30, 32, 32), Type: ${A^{88}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 14, 20, 24, 28, 30, 32, 32) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (8, 14, 18, 22, 24, 26, 28, 28), Type: ${A^{72}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 14, 18, 22, 24, 26, 28, 28) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (8, 14, 18, 22, 24, 26, 26, 26), Type: ${A^{70}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 14, 18, 22, 24, 26, 26, 26) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (8, 14, 18, 20, 22, 24, 24, 24), Type: ${A^{64}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 14, 18, 20, 22, 24, 24, 24) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (8, 14, 18, 20, 22, 23, 24, 24), Type: ${A^{63}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 14, 18, 20, 22, 23, 24, 24) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (8, 14, 18, 20, 22, 22, 22, 22), Type: ${A^{62}}_{1}$.
It turns out that in the current case of Cartan element h = (8, 14, 18, 20, 22, 22, 22, 22) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 16, 20, 22, 24, 26, 26), Type: ${A^{58}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 12, 16, 20, 22, 24, 26, 26) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 18, 20, 22, 24, 24), Type: ${A^{48}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 18, 20, 22, 24, 24) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 20, 20, 20), Type: ${A^{40}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 20, 20, 20) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 19, 20, 20), Type: ${A^{39}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 19, 20, 20) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 18, 18, 18), Type: ${A^{38}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 18, 18, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 12, 14, 16, 18, 18, 18), Type: ${A^{34}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 12, 14, 16, 18, 18, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 12, 14, 16, 17, 18, 18), Type: ${A^{33}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 12, 14, 16, 17, 18, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 12, 14, 16, 16, 16, 16), Type: ${A^{32}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 12, 14, 16, 16, 16, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 12, 14, 15, 16, 16, 16), Type: ${A^{31}}_{1}$.
It turns out that in the current case of Cartan element h = (6, 10, 12, 14, 15, 16, 16, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 8, 12, 14, 16, 18, 18, 18), Type: ${A^{30}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 8, 12, 14, 16, 18, 18, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 8, 10, 12, 14, 16, 16, 16), Type: ${A^{24}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 8, 10, 12, 14, 16, 16, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 8, 10, 12, 14, 15, 16, 16), Type: ${A^{23}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 8, 10, 12, 14, 15, 16, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 8, 10, 12, 14, 14, 14, 14), Type: ${A^{22}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 8, 10, 12, 14, 14, 14, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 7, 10, 12, 14, 15, 16, 16), Type: ${A^{22}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 7, 10, 12, 14, 15, 16, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 10, 12, 14, 14, 14), Type: ${A^{18}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 10, 12, 14, 14, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 10, 12, 12, 12, 12), Type: ${A^{16}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 10, 12, 12, 12, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 10, 11, 12, 12, 12), Type: ${A^{15}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 10, 11, 12, 12, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (3, 6, 8, 10, 11, 12, 12, 12), Type: ${A^{14}}_{1}$.
It turns out that in the current case of Cartan element h = (3, 6, 8, 10, 11, 12, 12, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 10, 10, 10, 10, 10), Type: ${A^{14}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 10, 10, 10, 10, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 9, 10, 11, 12, 12), Type: ${A^{14}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 9, 10, 11, 12, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (3, 6, 8, 9, 10, 11, 12, 12), Type: ${A^{13}}_{1}$.
It turns out that in the current case of Cartan element h = (3, 6, 8, 9, 10, 11, 12, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 9, 10, 10, 10, 10), Type: ${A^{13}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 9, 10, 10, 10, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (4, 6, 8, 8, 8, 8, 8, 8), Type: ${A^{12}}_{1}$.
It turns out that in the current case of Cartan element h = (4, 6, 8, 8, 8, 8, 8, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 10, 10, 10), Type: ${A^{10}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 10, 10, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 8, 8, 8, 8), Type: ${A^{8}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 8, 8, 8, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 7, 8, 8, 8, 8), Type: ${A^{7}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 7, 8, 8, 8, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 5, 6, 7, 8, 8, 8), Type: ${A^{6}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 5, 6, 7, 8, 8, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 6, 6, 6, 6, 6), Type: ${A^{6}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 6, 6, 6, 6, 6, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 3, 4, 5, 6, 7, 8, 8), Type: ${A^{5}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 3, 4, 5, 6, 7, 8, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 5, 6, 6, 6, 6, 6), Type: ${A^{5}}_{1}$.
It turns out that in the current case of Cartan element h = (2, 4, 5, 6, 6, 6, 6, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

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Extensions of the rationals used (5 total): $\mathbb{Q}$$\mathbb{Q}$, $\(\mathbb Q[\sqrt{2}]\)$, $\mathbb{Q}[\sqrt{-1},\sqrt{3}]$$\mathbb{Q} [\sqrt{-1},\sqrt{3}]$, $\mathbb{Q}[\sqrt{-1},\sqrt{2}]$$\mathbb{Q} [\sqrt{-1},\sqrt{2}]$, $\(\mathbb Q[\sqrt{-1}]\)$

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$A^{408}_1$
h-characteristic: (2, 2, 2, 2, 2, 2, 2, 2)
Length of the weight dual to h: 816
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $B^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{30\omega_{1}}+V_{26\omega_{1}}+V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&72h_{8}+70h_{7}+66h_{6}+60h_{5}+52h_{4}+42h_{3}+30h_{2}+16h_{1}\\ e&=&36g_{8}+70g_{7}+66g_{6}+60g_{5}+52g_{4}+42g_{3}+30g_{2}+16g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}+g_{-8}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&72h_{8}+70h_{7}+66h_{6}+60h_{5}+52h_{4}+42h_{3}+30h_{2}+16h_{1}\\ [h, e]&=&72g_{8}+140g_{7}+132g_{6}+120g_{5}+104g_{4}+84g_{3}+60g_{2}+32g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-7}-2g_{-8}\end{array}$
Centralizer type: $0$
Unfold the hidden panel for more information.

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$A^{280}_1$
h-characteristic: (2, 2, 2, 2, 2, 2, 2, 0)
Length of the weight dual to h: 560
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_8$ Containing regular semisimple subalgebra number 2: $B^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{26\omega_{1}}+V_{22\omega_{1}}+V_{18\omega_{1}}+3V_{14\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&56h_{8}+56h_{7}+54h_{6}+50h_{5}+44h_{4}+36h_{3}+26h_{2}+14h_{1}\\ e&=&28g_{22}+28g_{7}+54g_{6}+50g_{5}+44g_{4}+36g_{3}+26g_{2}+14g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&56h_{8}+56h_{7}+54h_{6}+50h_{5}+44h_{4}+36h_{3}+26h_{2}+14h_{1}\\ [h, e]&=&56g_{22}+56g_{7}+108g_{6}+100g_{5}+88g_{4}+72g_{3}+52g_{2}+28g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-7}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{184}_1$
h-characteristic: (2, 2, 2, 2, 2, 0, 2, 0)
Length of the weight dual to h: 368
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $D^{1}_7+A^{2}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_8$ Containing regular semisimple subalgebra number 4: $D^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+2V_{14\omega_{1}}+2V_{12\omega_{1}}+2V_{10\omega_{1}}+V_{6\omega_{1}}+3V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&44h_{8}+44h_{7}+42h_{6}+40h_{5}+36h_{4}+30h_{3}+22h_{2}+12h_{1}\\ e&=&21g_{22}+g_{21}+40g_{13}+21g_{7}+36g_{4}+30g_{3}+22g_{2}+12g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-7}+g_{-13}+g_{-21}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&44h_{8}+44h_{7}+42h_{6}+40h_{5}+36h_{4}+30h_{3}+22h_{2}+12h_{1}\\ [h, e]&=&42g_{22}+2g_{21}+80g_{13}+42g_{7}+72g_{4}+60g_{3}+44g_{2}+24g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-7}-2g_{-13}-2g_{-21}-2g_{-22}\end{array}$
Centralizer type: $0$
Unfold the hidden panel for more information.

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$A^{183}_1$
h-characteristic: (2, 2, 2, 2, 2, 1, 0, 1)
Length of the weight dual to h: 366
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $B^{1}_6+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{13\omega_{1}}+2V_{11\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+2V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&44h_{8}+43h_{7}+42h_{6}+40h_{5}+36h_{4}+30h_{3}+22h_{2}+12h_{1}\\ e&=&g_{22}+21g_{21}+40g_{5}+36g_{4}+30g_{3}+22g_{2}+12g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-21}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&44h_{8}+43h_{7}+42h_{6}+40h_{5}+36h_{4}+30h_{3}+22h_{2}+12h_{1}\\ [h, e]&=&2g_{22}+42g_{21}+80g_{5}+72g_{4}+60g_{3}+44g_{2}+24g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-21}-2g_{-22}\end{array}$
Centralizer type: $A_1$
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$A^{182}_1$
h-characteristic: (2, 2, 2, 2, 2, 2, 0, 0)
Length of the weight dual to h: 364
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_7$ Containing regular semisimple subalgebra number 2: $B^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{22\omega_{1}}+V_{18\omega_{1}}+V_{14\omega_{1}}+4V_{12\omega_{1}}+V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&42h_{8}+42h_{7}+42h_{6}+40h_{5}+36h_{4}+30h_{3}+22h_{2}+12h_{1}\\ e&=&21g_{34}+21g_{6}+40g_{5}+36g_{4}+30g_{3}+22g_{2}+12g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&42h_{8}+42h_{7}+42h_{6}+40h_{5}+36h_{4}+30h_{3}+22h_{2}+12h_{1}\\ [h, e]&=&42g_{34}+42g_{6}+80g_{5}+72g_{4}+60g_{3}+44g_{2}+24g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-34}\end{array}$
Centralizer type: $2A_1$
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$A^{120}_1$
h-characteristic: (2, 2, 2, 0, 2, 0, 2, 0)
Length of the weight dual to h: 240
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $D^{1}_6+B^{1}_2$ Containing regular semisimple subalgebra number 2: $B^{1}_5+A^{1}_3$ Containing regular semisimple subalgebra number 3: $B^{1}_8$ Containing regular semisimple subalgebra number 4: $D^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+2V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+V_{8\omega_{1}}+3V_{6\omega_{1}}+V_{4\omega_{1}}+2V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&36h_{8}+36h_{7}+34h_{6}+32h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ e&=&15g_{22}+3g_{21}+28g_{13}+4g_{12}+24g_{11}+15g_{7}+18g_{2}+10g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-7}+g_{-11}+g_{-12}+g_{-13}+g_{-21}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&36h_{8}+36h_{7}+34h_{6}+32h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ [h, e]&=&30g_{22}+6g_{21}+56g_{13}+8g_{12}+48g_{11}+30g_{7}+36g_{2}+20g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-7}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-21}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{114}_1$
h-characteristic: (2, 2, 2, 2, 0, 0, 2, 0)
Length of the weight dual to h: 228
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $B^{1}_5+A^{1}_2$ Containing regular semisimple subalgebra number 2: $D^{1}_7+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{12\omega_{1}}+3V_{10\omega_{1}}+2V_{8\omega_{1}}+V_{6\omega_{1}}+V_{4\omega_{1}}+4V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&34h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ e&=&2g_{33}+28g_{19}+15g_{15}+2g_{14}+24g_{3}+18g_{2}+10g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-14}+g_{-15}+g_{-19}+g_{-33}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&34h_{8}+34h_{7}+32h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ [h, e]&=&4g_{33}+56g_{19}+30g_{15}+4g_{14}+48g_{3}+36g_{2}+20g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-14}-2g_{-15}-2g_{-19}-2g_{-33}\end{array}$
Centralizer type: $0$
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$A^{112}_1$
h-characteristic: (2, 2, 2, 2, 0, 2, 0, 0)
Length of the weight dual to h: 224
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_6+A^{2}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_7$ Containing regular semisimple subalgebra number 5: $D^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+V_{12\omega_{1}}+5V_{10\omega_{1}}+V_{8\omega_{1}}+V_{6\omega_{1}}+5V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&32h_{8}+32h_{7}+32h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ e&=&15g_{34}+g_{33}+g_{20}+28g_{12}+15g_{6}+24g_{3}+18g_{2}+10g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-6}+g_{-12}+g_{-20}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&32h_{8}+32h_{7}+32h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ [h, e]&=&30g_{34}+2g_{33}+2g_{20}+56g_{12}+30g_{6}+48g_{3}+36g_{2}+20g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-6}-2g_{-12}-2g_{-20}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{2}_1$
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$A^{111}_1$
h-characteristic: (2, 2, 2, 2, 1, 0, 1, 0)
Length of the weight dual to h: 222
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_6+A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_5+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+2V_{11\omega_{1}}+3V_{10\omega_{1}}+2V_{9\omega_{1}}+V_{6\omega_{1}}+2V_{2\omega_{1}}+4V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&32h_{8}+32h_{7}+31h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ e&=&g_{34}+15g_{33}+15g_{20}+28g_{4}+24g_{3}+18g_{2}+10g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-20}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&32h_{8}+32h_{7}+31h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ [h, e]&=&2g_{34}+30g_{33}+30g_{20}+56g_{4}+48g_{3}+36g_{2}+20g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-20}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{110}_1$
h-characteristic: (2, 2, 2, 2, 2, 0, 0, 0)
Length of the weight dual to h: 220
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_6$ Containing regular semisimple subalgebra number 2: $B^{1}_5$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{18\omega_{1}}+V_{14\omega_{1}}+7V_{10\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+15V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&30h_{8}+30h_{7}+30h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ e&=&15g_{44}+15g_{5}+28g_{4}+24g_{3}+18g_{2}+10g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&30h_{8}+30h_{7}+30h_{6}+30h_{5}+28h_{4}+24h_{3}+18h_{2}+10h_{1}\\ [h, e]&=&30g_{44}+30g_{5}+56g_{4}+48g_{3}+36g_{2}+20g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-44}\end{array}$
Centralizer type: $A_3$
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$A^{88}_1$
h-characteristic: (2, 0, 2, 0, 2, 0, 2, 0)
Length of the weight dual to h: 176
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $D^{1}_5+B^{1}_3$ Containing regular semisimple subalgebra number 2: $B^{1}_4+D^{1}_4$ Containing regular semisimple subalgebra number 3: $B^{1}_8$ Containing regular semisimple subalgebra number 4: $D^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+2V_{8\omega_{1}}+4V_{6\omega_{1}}+V_{4\omega_{1}}+3V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&32h_{8}+32h_{7}+30h_{6}+28h_{5}+24h_{4}+20h_{3}+14h_{2}+8h_{1}\\ e&=&10g_{22}+6g_{21}+18g_{13}+10g_{12}+14g_{11}+6g_{10}+8g_{9}+10g_{7}\\ f&=&g_{-7}+g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-21}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&32h_{8}+32h_{7}+30h_{6}+28h_{5}+24h_{4}+20h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&20g_{22}+12g_{21}+36g_{13}+20g_{12}+28g_{11}+12g_{10}+16g_{9}+20g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-21}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{84}_1$
h-characteristic: (0, 2, 0, 2, 0, 2, 0, 1)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+3V_{12\omega_{1}}+V_{10\omega_{1}}+3V_{8\omega_{1}}+2V_{7\omega_{1}}+V_{6\omega_{1}}+3V_{4\omega_{1}}+V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&32h_{8}+31h_{7}+30h_{6}+27h_{5}+24h_{4}+19h_{3}+14h_{2}+7h_{1}\\ e&=&16g_{22}+7g_{16}+15g_{14}+15g_{13}+12g_{12}+12g_{11}+7g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-12}+g_{-13}+g_{-14}+g_{-16}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&32h_{8}+31h_{7}+30h_{6}+27h_{5}+24h_{4}+19h_{3}+14h_{2}+7h_{1}\\ [h, e]&=&32g_{22}+14g_{16}+30g_{14}+30g_{13}+24g_{12}+24g_{11}+14g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-14}-2g_{-16}-2g_{-22}\end{array}$
Centralizer type: $A^{4}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 3): $h_{7}+h_{5}+h_{3}-h_{1}$, $g_{1}+g_{-3}+g_{-5}+g_{-7}$, $g_{7}+g_{5}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{7}+h_{5}+h_{3}-h_{1}$
Cartan of centralizer (dimension: 1): $h_{7}+h_{5}+h_{3}-h_{1}$
Cartan-generating semisimple element: $h_{7}+h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-4x$
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{4}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/60\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{7}+h_{5}+h_{3}-h_{1}$
matching e: $g_{7}+g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{5}+h_{3}-h_{1}$
matching e: $g_{7}+g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}240\\ \end{pmatrix}$
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$A^{72}_1$
h-characteristic: (2, 2, 0, 2, 0, 0, 2, 0)
Length of the weight dual to h: 144
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $D^{1}_7+A^{2}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_6+B^{1}_2$ Containing regular semisimple subalgebra number 3: $B^{1}_4+D^{1}_4$ Containing regular semisimple subalgebra number 4: $B^{1}_8$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{12\omega_{1}}+3V_{10\omega_{1}}+2V_{8\omega_{1}}+5V_{6\omega_{1}}+2V_{4\omega_{1}}+4V_{2\omega_{1}}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&28h_{8}+28h_{7}+26h_{6}+24h_{5}+22h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&23/10g_{28}-1/3g_{27}+20/3g_{25}+88/5g_{22}-13/6g_{19}-8/3g_{18}+11/5g_{14}+8/3g_{12}-22/3g_{10}+32/5g_{7}+5/3g_{2}+8g_{1}\\ f&=&g_{-1}+4g_{-2}+g_{-7}-g_{-10}+g_{-12}+3g_{-14}+g_{-18}-4g_{-19}+g_{-22}+2g_{-25}-3g_{-27}-2g_{-28}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&28h_{8}+28h_{7}+26h_{6}+24h_{5}+22h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&23/5g_{28}-2/3g_{27}+40/3g_{25}+176/5g_{22}-13/3g_{19}-16/3g_{18}+22/5g_{14}+16/3g_{12}-44/3g_{10}+64/5g_{7}+10/3g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-8g_{-2}-2g_{-7}+2g_{-10}-2g_{-12}-6g_{-14}-2g_{-18}+8g_{-19}-2g_{-22}-4g_{-25}+6g_{-27}+4g_{-28}\end{array}$
Centralizer type: $0$
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$A^{70}_1$
h-characteristic: (2, 2, 1, 0, 1, 1, 0, 1)
Length of the weight dual to h: 140
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $B^{1}_4+A^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+2V_{11\omega_{1}}+V_{10\omega_{1}}+2V_{9\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&28h_{8}+27h_{7}+26h_{6}+24h_{5}+21h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&3g_{26}+4g_{22}+10g_{21}+18g_{18}+3g_{13}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-13}+g_{-18}+g_{-21}+g_{-22}+g_{-26}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&28h_{8}+27h_{7}+26h_{6}+24h_{5}+21h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&6g_{26}+8g_{22}+20g_{21}+36g_{18}+6g_{13}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-13}-2g_{-18}-2g_{-21}-2g_{-22}-2g_{-26}\end{array}$
Centralizer type: $A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 3): $h_{7}-h_{4}$, $g_{4}+g_{-7}$, $g_{7}+g_{-4}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{7}-h_{4}$
Cartan of centralizer (dimension: 1): $h_{7}-h_{4}$
Cartan-generating semisimple element: $h_{7}-h_{4}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-4x$
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{2}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{7}-h_{4}$
matching e: $g_{7}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}-h_{4}$
matching e: $g_{7}+g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
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$A^{70}_1$
h-characteristic: (2, 2, 0, 2, 0, 2, 0, 0)
Length of the weight dual to h: 140
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_5+A^{1}_3$ Containing regular semisimple subalgebra number 2: $D^{1}_5+B^{1}_2$ Containing regular semisimple subalgebra number 3: $B^{1}_4+A^{1}_3$ Containing regular semisimple subalgebra number 4: $B^{1}_7$ Containing regular semisimple subalgebra number 5: $D^{1}_7$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{12\omega_{1}}+2V_{10\omega_{1}}+4V_{8\omega_{1}}+3V_{6\omega_{1}}+4V_{4\omega_{1}}+2V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&26h_{8}+26h_{7}+26h_{6}+24h_{5}+22h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&10g_{34}+3g_{33}+3g_{20}+18g_{12}+4g_{11}+14g_{10}+10g_{6}+8g_{1}\\ f&=&g_{-1}+g_{-6}+g_{-10}+g_{-11}+g_{-12}+g_{-20}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&26h_{8}+26h_{7}+26h_{6}+24h_{5}+22h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&20g_{34}+6g_{33}+6g_{20}+36g_{12}+8g_{11}+28g_{10}+20g_{6}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-6}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-20}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{2}_1$
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$A^{64}_1$
h-characteristic: (2, 2, 2, 0, 0, 2, 0, 0)
Length of the weight dual to h: 128
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: $D^{1}_5+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_4+4A^{1}_1$ Containing regular semisimple subalgebra number 3: $D^{1}_5+A^{1}_2$ Containing regular semisimple subalgebra number 4: $B^{1}_4+A^{1}_2$ Containing regular semisimple subalgebra number 5: $B^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 6: $D^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 7: $D^{1}_6+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+3V_{10\omega_{1}}+4V_{8\omega_{1}}+3V_{6\omega_{1}}+V_{4\omega_{1}}+8V_{2\omega_{1}}+2V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&10g_{34}+g_{33}+g_{32}+g_{20}+18g_{18}+10g_{6}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-6}+g_{-18}+g_{-20}+g_{-32}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&20g_{34}+2g_{33}+2g_{32}+2g_{20}+36g_{18}+20g_{6}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-6}-2g_{-18}-2g_{-20}-2g_{-32}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{63}_1$
h-characteristic: (2, 2, 2, 0, 1, 0, 1, 0)
Length of the weight dual to h: 126
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $D^{1}_5+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_6+A^{1}_1$ Containing regular semisimple subalgebra number 4: $D^{1}_6+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+2V_{10\omega_{1}}+2V_{9\omega_{1}}+2V_{8\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{3\omega_{1}}+4V_{2\omega_{1}}+4V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+24h_{7}+23h_{6}+22h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&g_{34}+10g_{33}+g_{32}+10g_{20}+18g_{11}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-11}+g_{-20}+g_{-32}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+24h_{7}+23h_{6}+22h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&2g_{34}+20g_{33}+2g_{32}+20g_{20}+36g_{11}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-11}-2g_{-20}-2g_{-32}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{62}_1$
h-characteristic: (2, 2, 2, 1, 0, 0, 0, 1)
Length of the weight dual to h: 124
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $B^{1}_4+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+4V_{9\omega_{1}}+4V_{7\omega_{1}}+V_{6\omega_{1}}+7V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+23h_{7}+22h_{6}+21h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&g_{44}+10g_{32}+g_{22}+18g_{3}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-22}+g_{-32}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+23h_{7}+22h_{6}+21h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&2g_{44}+20g_{32}+2g_{22}+36g_{3}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-22}-2g_{-32}-2g_{-44}\end{array}$
Centralizer type: $B_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 10): $g_{-7}$, $g_{-5}$, $h_{5}$, $h_{7}$, $g_{5}$, $g_{6}+g_{-20}$, $g_{7}$, $g_{13}-g_{-14}$, $g_{14}-g_{-13}$, $g_{20}+g_{-6}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{7}$, $-h_{5}$
Cartan of centralizer (dimension: 2): $-h_{5}$, $-h_{7}$
Cartan-generating semisimple element: $-h_{7}-9h_{5}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{10}-492x^8+61488x^6-2311744x^4+8294400x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -18)(x -10)(x -8)(x -2)(x +2)(x +8)(x +10)(x +18)
Eigenvalues of ad H: $0$, $18$, $10$, $8$, $2$, $-2$, $-8$, $-10$, $-18$
10 eigenvectors of ad H: $(0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0)$, $(1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,1,0,0,0,0,0)$
Centralizer type: B^{1}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{7}-h_{5}$
matching e: $g_{14}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}-h_{5}$
matching e: $g_{14}-g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$
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$A^{62}_1$
h-characteristic: (2, 2, 2, 0, 2, 0, 0, 0)
Length of the weight dual to h: 124
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_5+A^{2}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $D^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_6$ Containing regular semisimple subalgebra number 5: $D^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+2V_{10\omega_{1}}+6V_{8\omega_{1}}+2V_{6\omega_{1}}+7V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&22h_{8}+22h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&10g_{44}+g_{32}+18g_{11}+10g_{5}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-5}+g_{-11}+g_{-32}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&22h_{8}+22h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&20g_{44}+2g_{32}+36g_{11}+20g_{5}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-5}-2g_{-11}-2g_{-32}-2g_{-44}\end{array}$
Centralizer type: $B_2$
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$A^{61}_1$
h-characteristic: (2, 2, 2, 1, 0, 1, 0, 0)
Length of the weight dual to h: 122
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_5+A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_4+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+2V_{9\omega_{1}}+4V_{8\omega_{1}}+2V_{7\omega_{1}}+V_{6\omega_{1}}+2V_{2\omega_{1}}+8V_{\omega_{1}}+9V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&22h_{8}+22h_{7}+22h_{6}+21h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&g_{44}+10g_{43}+10g_{19}+18g_{3}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-19}+g_{-43}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&22h_{8}+22h_{7}+22h_{6}+21h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&2g_{44}+20g_{43}+20g_{19}+36g_{3}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-19}-2g_{-43}-2g_{-44}\end{array}$
Centralizer type: $3A_1$
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$A^{60}_1$
h-characteristic: (2, 2, 2, 2, 0, 0, 0, 0)
Length of the weight dual to h: 120
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_5$ Containing regular semisimple subalgebra number 2: $B^{1}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{14\omega_{1}}+V_{10\omega_{1}}+8V_{8\omega_{1}}+V_{6\omega_{1}}+V_{2\omega_{1}}+28V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+20h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ e&=&10g_{52}+10g_{4}+18g_{3}+14g_{2}+8g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+20h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+14h_{2}+8h_{1}\\ [h, e]&=&20g_{52}+20g_{4}+36g_{3}+28g_{2}+16g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-52}\end{array}$
Centralizer type: $D_4$
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$A^{58}_1$
h-characteristic: (0, 2, 0, 2, 0, 0, 2, 0)
Length of the weight dual to h: 116
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_6+A^{2}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_5+B^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{12\omega_{1}}+3V_{10\omega_{1}}+3V_{8\omega_{1}}+5V_{6\omega_{1}}+3V_{4\omega_{1}}+4V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&26h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&12g_{22}+g_{21}+12g_{20}+10g_{19}+6g_{16}+10g_{11}+6g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-16}+g_{-19}+g_{-20}+g_{-21}+g_{-22}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&26h_{8}+26h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&24g_{22}+2g_{21}+24g_{20}+20g_{19}+12g_{16}+20g_{11}+12g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-16}-2g_{-19}-2g_{-20}-2g_{-21}-2g_{-22}\end{array}$
Centralizer type: $0$
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$A^{56}_1$
h-characteristic: (0, 2, 0, 2, 0, 2, 0, 0)
Length of the weight dual to h: 112
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2D^{1}_4$ Containing regular semisimple subalgebra number 2: $D^{1}_4+B^{1}_3$ Containing regular semisimple subalgebra number 3: $A^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{12\omega_{1}}+3V_{10\omega_{1}}+V_{8\omega_{1}}+9V_{6\omega_{1}}+V_{4\omega_{1}}+3V_{2\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&6g_{34}+6g_{33}+6g_{20}+6g_{16}+10g_{12}+10g_{11}+6g_{6}+6g_{2}\\ f&=&g_{-2}+g_{-6}+g_{-11}+g_{-12}+g_{-16}+g_{-20}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+24h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&12g_{34}+12g_{33}+12g_{20}+12g_{16}+20g_{12}+20g_{11}+12g_{6}+12g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-6}-2g_{-11}-2g_{-12}-2g_{-16}-2g_{-20}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{2}_1$
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$A^{48}_1$
h-characteristic: (2, 0, 0, 2, 0, 0, 2, 0)
Length of the weight dual to h: 96
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_4+B^{1}_3$ Containing regular semisimple subalgebra number 2: $D^{1}_6+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{10\omega_{1}}+3V_{8\omega_{1}}+6V_{6\omega_{1}}+3V_{4\omega_{1}}+6V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+24h_{7}+22h_{6}+20h_{5}+18h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{30}+6g_{22}+6g_{21}+6g_{20}+6g_{16}+10g_{12}+4g_{11}\\ f&=&g_{-11}+g_{-12}+g_{-16}+g_{-20}+g_{-21}+g_{-22}+g_{-30}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+24h_{7}+22h_{6}+20h_{5}+18h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{30}+12g_{22}+12g_{21}+12g_{20}+12g_{16}+20g_{12}+8g_{11}\\ [h, f]&=&-2g_{-11}-2g_{-12}-2g_{-16}-2g_{-20}-2g_{-21}-2g_{-22}-2g_{-30}\end{array}$
Centralizer type: $0$
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$A^{45}_1$
h-characteristic: (0, 1, 1, 0, 1, 1, 0, 1)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_5+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{9\omega_{1}}+3V_{8\omega_{1}}+2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{3\omega_{1}}+2V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&24h_{8}+23h_{7}+22h_{6}+20h_{5}+17h_{4}+14h_{3}+10h_{2}+5h_{1}\\ e&=&5g_{23}+9g_{22}+3g_{21}+8g_{20}+8g_{19}+4g_{18}+5g_{10}\\ f&=&g_{-10}+g_{-18}+g_{-19}+g_{-20}+g_{-21}+g_{-22}+g_{-23}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&24h_{8}+23h_{7}+22h_{6}+20h_{5}+17h_{4}+14h_{3}+10h_{2}+5h_{1}\\ [h, e]&=&10g_{23}+18g_{22}+6g_{21}+16g_{20}+16g_{19}+8g_{18}+10g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-21}-2g_{-22}-2g_{-23}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 3): $h_{7}+h_{4}-h_{1}$, $g_{1}+g_{-4}+g_{-7}$, $g_{7}+g_{4}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{7}+h_{4}-h_{1}$
Cartan of centralizer (dimension: 1): $h_{7}+h_{4}-h_{1}$
Cartan-generating semisimple element: $h_{7}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-4x$
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{7}+h_{4}-h_{1}$
matching e: $g_{7}+g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{4}-h_{1}$
matching e: $g_{7}+g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
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$A^{40}_1$
h-characteristic: (2, 0, 2, 0, 0, 2, 0, 0)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 14
Containing regular semisimple subalgebra number 1: $D^{1}_4+A^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_4+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_3+A^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $D^{1}_8$ Containing regular semisimple subalgebra number 5: $B^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 6: $D^{1}_6+2A^{1}_1$ Containing regular semisimple subalgebra number 7: $B^{1}_5+A^{1}_3$ Containing regular semisimple subalgebra number 8: $D^{1}_5+A^{1}_3$ Containing regular semisimple subalgebra number 9: $B^{1}_4+D^{1}_4$ Containing regular semisimple subalgebra number 10: $2D^{1}_4$ Containing regular semisimple subalgebra number 11: $B^{1}_7$ Containing regular semisimple subalgebra number 12: $D^{1}_6+A^{2}_1$ Containing regular semisimple subalgebra number 13: $D^{1}_5+B^{1}_2$ Containing regular semisimple subalgebra number 14: $D^{1}_4+B^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{10\omega_{1}}+2V_{8\omega_{1}}+7V_{6\omega_{1}}+5V_{4\omega_{1}}+7V_{2\omega_{1}}+V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&3g_{38}+6g_{34}+g_{27}+3g_{26}+10g_{18}+4g_{10}+6g_{9}+6g_{6}\\ f&=&g_{-6}+g_{-9}+g_{-10}+g_{-18}+g_{-26}+g_{-27}+g_{-34}+g_{-38}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+20h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&6g_{38}+12g_{34}+2g_{27}+6g_{26}+20g_{18}+8g_{10}+12g_{9}+12g_{6}\\ [h, f]&=&-2g_{-6}-2g_{-9}-2g_{-10}-2g_{-18}-2g_{-26}-2g_{-27}-2g_{-34}-2g_{-38}\end{array}$
Centralizer type: $0$
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$A^{39}_1$
h-characteristic: (2, 0, 2, 0, 1, 0, 1, 0)
Length of the weight dual to h: 78
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $D^{1}_4+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_3+A^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_6+A^{1}_1$ Containing regular semisimple subalgebra number 4: $D^{1}_6+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{10\omega_{1}}+V_{8\omega_{1}}+2V_{7\omega_{1}}+4V_{6\omega_{1}}+4V_{5\omega_{1}}+2V_{4\omega_{1}}+2V_{3\omega_{1}}+4V_{2\omega_{1}}+2V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+20h_{7}+19h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&g_{34}+6g_{33}+3g_{32}+6g_{20}+10g_{11}+4g_{10}+6g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-11}+g_{-20}+g_{-32}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+20h_{7}+19h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&2g_{34}+12g_{33}+6g_{32}+12g_{20}+20g_{11}+8g_{10}+12g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-20}-2g_{-32}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{38}_1$
h-characteristic: (2, 1, 0, 1, 1, 0, 1, 0)
Length of the weight dual to h: 76
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_4+A^{1}_3$ Containing regular semisimple subalgebra number 2: $B^{1}_3+A^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{9\omega_{1}}+2V_{7\omega_{1}}+4V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+6V_{3\omega_{1}}+2V_{2\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+20h_{7}+19h_{6}+18h_{5}+16h_{4}+13h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{34}+6g_{33}+3g_{25}+6g_{20}+10g_{17}+3g_{12}+6g_{1}\\ f&=&g_{-1}+g_{-12}+g_{-17}+g_{-20}+g_{-25}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+20h_{7}+19h_{6}+18h_{5}+16h_{4}+13h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{34}+12g_{33}+6g_{25}+12g_{20}+20g_{17}+6g_{12}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-12}-2g_{-17}-2g_{-20}-2g_{-25}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 4): $h_{6}-h_{3}$, $g_{3}+g_{-6}$, $g_{6}+g_{-3}$, $g_{8}+g_{-8}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{6}-h_{3}$
Cartan of centralizer (dimension: 2): $h_{6}-h_{3}$, $g_{8}+g_{-8}$
Cartan-generating semisimple element: $h_{6}-h_{3}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^4-4x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
4 eigenvectors of ad H: $(1,0,0,0)$, $(0,0,0,1)$, $(0,0,1,0)$, $(0,1,0,0)$
Centralizer type: A^{2}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{6}-h_{3}$
matching e: $g_{6}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}-h_{3}$
matching e: $g_{6}+g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
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$A^{38}_1$
h-characteristic: (2, 0, 2, 0, 2, 0, 0, 0)
Length of the weight dual to h: 76
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_4+A^{1}_3$ Containing regular semisimple subalgebra number 2: $D^{1}_4+B^{1}_2$ Containing regular semisimple subalgebra number 3: $B^{1}_3+A^{1}_3$ Containing regular semisimple subalgebra number 4: $B^{1}_6$ Containing regular semisimple subalgebra number 5: $D^{1}_6$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{10\omega_{1}}+V_{8\omega_{1}}+8V_{6\omega_{1}}+6V_{4\omega_{1}}+3V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&6g_{44}+3g_{43}+3g_{19}+10g_{11}+4g_{10}+6g_{9}+6g_{5}\\ f&=&g_{-5}+g_{-9}+g_{-10}+g_{-11}+g_{-19}+g_{-43}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+18h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&12g_{44}+6g_{43}+6g_{19}+20g_{11}+8g_{10}+12g_{9}+12g_{5}\\ [h, f]&=&-2g_{-5}-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-19}-2g_{-43}-2g_{-44}\end{array}$
Centralizer type: $B_2$
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$A^{37}_1$
h-characteristic: (0, 2, 0, 1, 1, 0, 1, 0)
Length of the weight dual to h: 74
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_5+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+3V_{8\omega_{1}}+2V_{7\omega_{1}}+V_{6\omega_{1}}+6V_{5\omega_{1}}+3V_{4\omega_{1}}+2V_{3\omega_{1}}+4V_{2\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+20h_{7}+19h_{6}+18h_{5}+16h_{4}+13h_{3}+10h_{2}+5h_{1}\\ e&=&9g_{34}+g_{33}+g_{20}+8g_{19}+8g_{18}+5g_{16}+5g_{2}\\ f&=&g_{-2}+g_{-16}+g_{-18}+g_{-19}+g_{-20}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+20h_{7}+19h_{6}+18h_{5}+16h_{4}+13h_{3}+10h_{2}+5h_{1}\\ [h, e]&=&18g_{34}+2g_{33}+2g_{20}+16g_{19}+16g_{18}+10g_{16}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-16}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 4): $h_{6}+h_{3}-h_{1}$, $g_{1}+g_{-3}+g_{-6}$, $g_{6}+g_{3}+g_{-1}$, $g_{8}+g_{-8}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{6}+h_{3}-h_{1}$
Cartan of centralizer (dimension: 2): $h_{6}+h_{3}-h_{1}$, $g_{8}+g_{-8}$
Cartan-generating semisimple element: $h_{6}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^4-4x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
4 eigenvectors of ad H: $(1,0,0,0)$, $(0,0,0,1)$, $(0,0,1,0)$, $(0,1,0,0)$
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{6}+h_{3}-h_{1}$
matching e: $g_{6}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}+h_{3}-h_{1}$
matching e: $g_{6}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
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$A^{36}_1$
h-characteristic: (0, 2, 0, 2, 0, 0, 0, 1)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_5+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+3V_{8\omega_{1}}+5V_{6\omega_{1}}+2V_{5\omega_{1}}+7V_{4\omega_{1}}+2V_{2\omega_{1}}+2V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&20h_{8}+19h_{7}+18h_{6}+17h_{5}+16h_{4}+13h_{3}+10h_{2}+5h_{1}\\ e&=&g_{34}+9g_{33}+8g_{26}+5g_{16}+8g_{11}+5g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-16}+g_{-26}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&20h_{8}+19h_{7}+18h_{6}+17h_{5}+16h_{4}+13h_{3}+10h_{2}+5h_{1}\\ [h, e]&=&2g_{34}+18g_{33}+16g_{26}+10g_{16}+16g_{11}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-16}-2g_{-26}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 6): $g_{-6}$, $h_{6}$, $h_{7}+h_{5}+h_{3}-h_{1}$, $g_{1}+g_{-3}+g_{-20}$, $g_{6}$, $g_{20}+g_{3}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}+h_{5}+h_{3}-h_{1}$, $-h_{6}$
Cartan of centralizer (dimension: 2): $h_{7}+h_{5}+h_{3}-h_{1}$, $-h_{6}$
Cartan-generating semisimple element: $h_{7}-h_{6}+h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-20x^4+64x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -4)(x -2)(x +2)(x +4)
Eigenvalues of ad H: $0$, $4$, $2$, $-2$, $-4$
6 eigenvectors of ad H: $(0,1,0,0,0,0)$, $(0,0,1,0,0,0)$, $(1,0,0,0,0,0)$, $(0,0,0,0,0,1)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$
Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{7}+h_{6}+h_{5}+h_{3}-h_{1}$
matching e: $g_{20}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{6}+h_{5}+h_{3}-h_{1}$
matching e: $g_{20}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{35}_1$
h-characteristic: (0, 2, 0, 2, 0, 1, 0, 0)
Length of the weight dual to h: 70
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_5$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+3V_{8\omega_{1}}+V_{6\omega_{1}}+10V_{5\omega_{1}}+3V_{4\omega_{1}}+V_{2\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+18h_{7}+18h_{6}+17h_{5}+16h_{4}+13h_{3}+10h_{2}+5h_{1}\\ e&=&9g_{44}+5g_{16}+8g_{12}+8g_{11}+5g_{2}\\ f&=&g_{-2}+g_{-11}+g_{-12}+g_{-16}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+18h_{7}+18h_{6}+17h_{5}+16h_{4}+13h_{3}+10h_{2}+5h_{1}\\ [h, e]&=&18g_{44}+10g_{16}+16g_{12}+16g_{11}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-11}-2g_{-12}-2g_{-16}-2g_{-44}\end{array}$
Centralizer type: $B_2+A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 13): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{5}+h_{3}-h_{1}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-3}+g_{-5}$, $g_{5}+g_{3}+g_{-1}$, $g_{7}$, $g_{8}$, $g_{15}$, $g_{22}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{5}+h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 3): $-h_{7}$, $h_{5}+h_{3}-h_{1}$, $-h_{8}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-919x^{11}+245103x^9-21686557x^7+620380996x^5-2149991424x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -23)(x -16)(x -9)(x -7)(x -2)(x +2)(x +7)(x +9)(x +16)(x +23)
Eigenvalues of ad H: $0$, $23$, $16$, $9$, $7$, $2$, $-2$, $-7$, $-9$, $-16$, $-23$
13 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15 & -1/30\\ -1/30 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -60\\ -60 & 120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{5}+h_{3}-h_{1}$
matching e: $g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}+h_{3}-h_{1}$
matching e: $g_{5}+g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{34}_1$
h-characteristic: (2, 2, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 68
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_4+A^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_3+A^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $D^{1}_5+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_5+A^{1}_2$ Containing regular semisimple subalgebra number 5: $D^{1}_5+A^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+3V_{8\omega_{1}}+5V_{6\omega_{1}}+6V_{4\omega_{1}}+10V_{2\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+18h_{7}+18h_{6}+16h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&2g_{42}+6g_{34}+g_{27}+2g_{26}+10g_{24}+6g_{6}+6g_{1}\\ f&=&g_{-1}+g_{-6}+g_{-24}+g_{-26}+g_{-27}+g_{-34}+g_{-42}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+18h_{7}+18h_{6}+16h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&4g_{42}+12g_{34}+2g_{27}+4g_{26}+20g_{24}+12g_{6}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-6}-2g_{-24}-2g_{-26}-2g_{-27}-2g_{-34}-2g_{-42}\end{array}$
Centralizer type: $A^{6}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 3): $h_{8}-h_{3}$, $g_{8}+2g_{4}-g_{-11}$, $g_{11}-1/2g_{-4}-1/2g_{-8}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{8}-h_{3}$
Cartan of centralizer (dimension: 1): $h_{8}-h_{3}$
Cartan-generating semisimple element: $h_{8}-h_{3}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-x$
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: $0$, $1$, $-1$
3 eigenvectors of ad H: $(1,0,0)$, $(0,1,0)$, $(0,0,1)$
Centralizer type: A^{6}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/90\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}-2h_{3}$
matching e: $g_{8}+2g_{4}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}-2h_{3}$
matching e: $g_{8}+2g_{4}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{33}_1$
h-characteristic: (2, 2, 0, 0, 1, 0, 1, 0)
Length of the weight dual to h: 66
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $B^{1}_3+A^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_5+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{8\omega_{1}}+2V_{7\omega_{1}}+3V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+4V_{3\omega_{1}}+5V_{2\omega_{1}}+4V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+18h_{7}+17h_{6}+16h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&2g_{42}+g_{34}+6g_{27}+2g_{26}+10g_{17}+6g_{1}\\ f&=&g_{-1}+g_{-17}+g_{-26}+g_{-27}+g_{-34}+g_{-42}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+18h_{7}+17h_{6}+16h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&4g_{42}+2g_{34}+12g_{27}+4g_{26}+20g_{17}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-17}-2g_{-26}-2g_{-27}-2g_{-34}-2g_{-42}\end{array}$
Centralizer type: $A_1$
Unfold the hidden panel for more information.

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$A^{32}_1$
h-characteristic: (2, 2, 0, 0, 2, 0, 0, 0)
Length of the weight dual to h: 64
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: $D^{1}_4+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_4+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_3+4A^{1}_1$ Containing regular semisimple subalgebra number 4: $D^{1}_4+A^{1}_2$ Containing regular semisimple subalgebra number 5: $B^{1}_3+A^{1}_2$ Containing regular semisimple subalgebra number 6: $B^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 7: $D^{1}_5+2A^{1}_1$ Containing regular semisimple subalgebra number 8: $D^{1}_5+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{8\omega_{1}}+7V_{6\omega_{1}}+3V_{4\omega_{1}}+12V_{2\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+16h_{7}+16h_{6}+16h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&g_{47}+6g_{44}+g_{38}+g_{26}+g_{25}+10g_{17}+6g_{5}+6g_{1}\\ f&=&g_{-1}+g_{-5}+g_{-17}+g_{-25}+g_{-26}+g_{-38}+g_{-44}+g_{-47}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+16h_{7}+16h_{6}+16h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&2g_{47}+12g_{44}+2g_{38}+2g_{26}+2g_{25}+20g_{17}+12g_{5}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-5}-2g_{-17}-2g_{-25}-2g_{-26}-2g_{-38}-2g_{-44}-2g_{-47}\end{array}$
Centralizer type: $2A_1$
Unfold the hidden panel for more information.

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$A^{31}_1$
h-characteristic: (2, 2, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 62
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_4+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_3+3A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_5+A^{1}_1$ Containing regular semisimple subalgebra number 5: $D^{1}_5+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+2V_{7\omega_{1}}+5V_{6\omega_{1}}+2V_{5\omega_{1}}+V_{4\omega_{1}}+2V_{3\omega_{1}}+6V_{2\omega_{1}}+8V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+16h_{7}+16h_{6}+15h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&g_{44}+6g_{43}+g_{42}+g_{31}+6g_{19}+10g_{10}+6g_{1}\\ f&=&g_{-1}+g_{-10}+g_{-19}+g_{-31}+g_{-42}+g_{-43}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+16h_{7}+16h_{6}+15h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&2g_{44}+12g_{43}+2g_{42}+2g_{31}+12g_{19}+20g_{10}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-10}-2g_{-19}-2g_{-31}-2g_{-42}-2g_{-43}-2g_{-44}\end{array}$
Centralizer type: $A^{2}_1+A_1$
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$A^{30}_1$
h-characteristic: (2, 2, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_3+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+4V_{7\omega_{1}}+3V_{6\omega_{1}}+4V_{5\omega_{1}}+7V_{2\omega_{1}}+8V_{\omega_{1}}+11V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+16h_{7}+15h_{6}+14h_{5}+13h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&g_{52}+6g_{42}+g_{34}+6g_{31}+10g_{2}+6g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-31}+g_{-34}+g_{-42}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+16h_{7}+15h_{6}+14h_{5}+13h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&2g_{52}+12g_{42}+2g_{34}+12g_{31}+20g_{2}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-31}-2g_{-34}-2g_{-42}-2g_{-52}\end{array}$
Centralizer type: $B_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 11): $g_{-6}$, $g_{-4}$, $h_{4}$, $h_{6}$, $g_{4}$, $g_{5}+g_{-19}$, $g_{6}$, $g_{8}+g_{-8}$, $g_{12}-g_{-13}$, $g_{13}-g_{-12}$, $g_{19}+g_{-5}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{6}$, $-h_{4}$
Cartan of centralizer (dimension: 3): $-h_{4}$, $-g_{8}-g_{-8}$, $-h_{6}$
Cartan-generating semisimple element: $-h_{6}-9h_{4}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{11}-492x^9+61488x^7-2311744x^5+8294400x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -18)(x -10)(x -8)(x -2)(x +2)(x +8)(x +10)(x +18)
Eigenvalues of ad H: $0$, $18$, $10$, $8$, $2$, $-2$, $-8$, $-10$, $-18$
11 eigenvectors of ad H: $(0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0)$, $(1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,1,0,0,0,0,0,0)$
Centralizer type: B^{1}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{6}-h_{4}$
matching e: $g_{13}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}-h_{4}$
matching e: $g_{13}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$
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$A^{30}_1$
h-characteristic: (2, 2, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $D^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_5$ Containing regular semisimple subalgebra number 3: $D^{1}_5$ Containing regular semisimple subalgebra number 4: $D^{1}_4+A^{2}_1$ Containing regular semisimple subalgebra number 5: $B^{1}_3+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+V_{8\omega_{1}}+9V_{6\omega_{1}}+V_{4\omega_{1}}+9V_{2\omega_{1}}+21V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+14h_{6}+14h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&6g_{52}+g_{51}+g_{18}+10g_{10}+6g_{4}+6g_{1}\\ f&=&g_{-1}+g_{-4}+g_{-10}+g_{-18}+g_{-51}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+14h_{6}+14h_{5}+14h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&12g_{52}+2g_{51}+2g_{18}+20g_{10}+12g_{4}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-4}-2g_{-10}-2g_{-18}-2g_{-51}-2g_{-52}\end{array}$
Centralizer type: $B_3$
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$A^{30}_1$
h-characteristic: (0, 0, 2, 0, 0, 2, 0, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{1}_4+B^{1}_2$ Containing regular semisimple subalgebra number 2: $2A^{1}_3+B^{1}_2$ Containing regular semisimple subalgebra number 3: $A^{1}_4+A^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{8\omega_{1}}+6V_{6\omega_{1}}+9V_{4\omega_{1}}+6V_{2\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&18h_{8}+18h_{7}+18h_{6}+16h_{5}+14h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&6g_{34}+4g_{29}+3g_{27}+6g_{19}+4g_{11}+4g_{10}\\ f&=&g_{-10}+g_{-11}+g_{-19}+g_{-27}+g_{-29}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&18h_{8}+18h_{7}+18h_{6}+16h_{5}+14h_{4}+12h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&12g_{34}+8g_{29}+6g_{27}+12g_{19}+8g_{11}+8g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-11}-2g_{-19}-2g_{-27}-2g_{-29}-2g_{-34}\end{array}$
Centralizer type: $A^{10}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 4): $h_{7}+h_{5}+h_{4}-h_{1}$, $h_{8}$, $g_{9}-1/2g_{-2}-1/2g_{-4}+g_{-5}-1/2g_{-15}$, $g_{15}-g_{5}+2g_{4}+2g_{2}-g_{-9}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{8}$, $h_{7}+h_{5}+h_{4}-h_{1}$
Cartan of centralizer (dimension: 2): $-h_{8}$, $h_{7}+h_{5}+h_{4}-h_{1}$
Cartan-generating semisimple element: $-h_{8}+9h_{7}+9h_{5}+9h_{4}-9h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -9 & 0\\ 0 & 0 & 0 & 9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^4-81x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -9)(x +9)
Eigenvalues of ad H: $0$, $9$, $-9$
4 eigenvectors of ad H: $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,0,1)$, $(0,0,1,0)$
Centralizer type: A^{10}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/150\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}+2h_{7}+2h_{5}+2h_{4}-2h_{1}$
matching e: $g_{15}-g_{5}+2g_{4}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}+2h_{7}+2h_{5}+2h_{4}-2h_{1}$
matching e: $g_{15}-g_{5}+2g_{4}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}600\\ \end{pmatrix}$
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$A^{29}_1$
h-characteristic: (2, 2, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 58
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_4+A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_3+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+2V_{7\omega_{1}}+7V_{6\omega_{1}}+2V_{5\omega_{1}}+2V_{2\omega_{1}}+12V_{\omega_{1}}+18V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+14h_{6}+14h_{5}+13h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&g_{52}+6g_{51}+6g_{18}+10g_{2}+6g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-18}+g_{-51}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+14h_{6}+14h_{5}+13h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&2g_{52}+12g_{51}+12g_{18}+20g_{2}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-18}-2g_{-51}-2g_{-52}\end{array}$
Centralizer type: $A_3+A_1$
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$A^{28}_1$
h-characteristic: (2, 2, 2, 0, 0, 0, 0, 0)
Length of the weight dual to h: 56
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $D^{1}_4$ Containing regular semisimple subalgebra number 2: $B^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{10\omega_{1}}+11V_{6\omega_{1}}+V_{2\omega_{1}}+45V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+12h_{3}+10h_{2}+6h_{1}\\ e&=&6g_{58}+6g_{3}+10g_{2}+6g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+12h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&12g_{58}+12g_{3}+20g_{2}+12g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-58}\end{array}$
Centralizer type: $D_5$
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$A^{24}_1$
h-characteristic: (0, 2, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 48
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: $A^{1}_4+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+B^{1}_2+A^{1}_2$ Containing regular semisimple subalgebra number 3: $A^{1}_4+A^{1}_2$ Containing regular semisimple subalgebra number 4: $B^{1}_4+D^{1}_4$ Containing regular semisimple subalgebra number 5: $2D^{1}_4$ Containing regular semisimple subalgebra number 6: $D^{1}_4+A^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 7: $D^{1}_4+B^{1}_2+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{8\omega_{1}}+7V_{6\omega_{1}}+8V_{4\omega_{1}}+12V_{2\omega_{1}}+2V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&6g_{34}+g_{33}+g_{32}+4g_{29}+6g_{25}+g_{20}+4g_{2}\\ f&=&g_{-2}+g_{-20}+g_{-25}+g_{-29}+g_{-32}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+16h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&12g_{34}+2g_{33}+2g_{32}+8g_{29}+12g_{25}+2g_{20}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-20}-2g_{-25}-2g_{-29}-2g_{-32}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $0$
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$A^{23}_1$
h-characteristic: (0, 2, 0, 0, 1, 0, 1, 0)
Length of the weight dual to h: 46
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_4+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_4+B^{1}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{8\omega_{1}}+5V_{6\omega_{1}}+4V_{5\omega_{1}}+3V_{4\omega_{1}}+6V_{3\omega_{1}}+7V_{2\omega_{1}}+2V_{\omega_{1}}+4V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+16h_{7}+15h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{34}+6g_{33}+g_{32}+6g_{31}+4g_{23}+4g_{2}\\ f&=&g_{-2}+g_{-23}+g_{-31}+g_{-32}+g_{-33}+g_{-34}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+16h_{7}+15h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{34}+12g_{33}+2g_{32}+12g_{31}+8g_{23}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-23}-2g_{-31}-2g_{-32}-2g_{-33}-2g_{-34}\end{array}$
Centralizer type: $A_1$
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$A^{22}_1$
h-characteristic: (1, 0, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 44
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $2A^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_4+A^{1}_3$ Containing regular semisimple subalgebra number 4: $D^{1}_4+A^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{7\omega_{1}}+3V_{6\omega_{1}}+4V_{5\omega_{1}}+5V_{4\omega_{1}}+6V_{3\omega_{1}}+5V_{2\omega_{1}}+4V_{\omega_{1}}+3V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+16h_{7}+15h_{6}+14h_{5}+12h_{4}+10h_{3}+7h_{2}+4h_{1}\\ e&=&3g_{38}+4g_{34}+3g_{30}+g_{27}+3g_{26}+3g_{18}+4g_{16}\\ f&=&g_{-16}+g_{-18}+g_{-26}+g_{-27}+g_{-30}+g_{-34}+g_{-38}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+16h_{7}+15h_{6}+14h_{5}+12h_{4}+10h_{3}+7h_{2}+4h_{1}\\ [h, e]&=&6g_{38}+8g_{34}+6g_{30}+2g_{27}+6g_{26}+6g_{18}+8g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-18}-2g_{-26}-2g_{-27}-2g_{-30}-2g_{-34}-2g_{-38}\end{array}$
Centralizer type: $A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 3): $h_{6}-h_{2}$, $g_{2}+g_{-6}$, $g_{6}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 1): $h_{6}-h_{2}$
Cartan of centralizer (dimension: 1): $h_{6}-h_{2}$
Cartan-generating semisimple element: $h_{6}-h_{2}$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^3-4x$
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
3 eigenvectors of ad H: $(1,0,0)$, $(0,0,1)$, $(0,1,0)$
Centralizer type: A^{2}_1
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{6}-h_{2}$
matching e: $g_{6}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}-h_{2}$
matching e: $g_{6}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{22}_1$
h-characteristic: (0, 2, 0, 0, 2, 0, 0, 0)
Length of the weight dual to h: 44
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: $2A^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $2A^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_3+B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 5: $A^{1}_4+A^{2}_1$ Containing regular semisimple subalgebra number 6: $B^{1}_4+A^{1}_3$ Containing regular semisimple subalgebra number 7: $D^{1}_4+A^{1}_3$ Containing regular semisimple subalgebra number 8: $D^{1}_4+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{8\omega_{1}}+5V_{6\omega_{1}}+11V_{4\omega_{1}}+10V_{2\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&3g_{51}+3g_{43}+g_{33}+g_{20}+3g_{19}+3g_{18}+4g_{10}+4g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-18}+g_{-19}+g_{-20}+g_{-33}+g_{-43}+g_{-51}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&6g_{51}+6g_{43}+2g_{33}+2g_{20}+6g_{19}+6g_{18}+8g_{10}+8g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-33}-2g_{-43}-2g_{-51}\end{array}$
Centralizer type: $2A_1$
Unfold the hidden panel for more information.

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$A^{21}_1$
h-characteristic: (1, 0, 1, 1, 0, 0, 0, 1)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_3+B^{1}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{7\omega_{1}}+2V_{6\omega_{1}}+4V_{5\omega_{1}}+7V_{4\omega_{1}}+4V_{3\omega_{1}}+7V_{2\omega_{1}}+2V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+15h_{7}+14h_{6}+13h_{5}+12h_{4}+10h_{3}+7h_{2}+4h_{1}\\ e&=&3g_{36}+g_{34}+4g_{33}+3g_{32}+4g_{16}+3g_{11}\\ f&=&g_{-11}+g_{-16}+g_{-32}+g_{-33}+g_{-34}+g_{-36}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+15h_{7}+14h_{6}+13h_{5}+12h_{4}+10h_{3}+7h_{2}+4h_{1}\\ [h, e]&=&6g_{36}+2g_{34}+8g_{33}+6g_{32}+8g_{16}+6g_{11}\\ [h, f]&=&-2g_{-11}-2g_{-16}-2g_{-32}-2g_{-33}-2g_{-34}-2g_{-36}\end{array}$
Centralizer type: $A^{2}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 6): $g_{-6}$, $h_{6}$, $h_{7}+h_{5}-h_{2}$, $g_{2}+g_{-20}$, $g_{6}$, $g_{20}+g_{-2}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{7}+h_{5}-h_{2}$, $-h_{6}$
Cartan of centralizer (dimension: 2): $h_{7}+h_{5}-h_{2}$, $-h_{6}$
Cartan-generating semisimple element: $h_{7}-h_{6}+h_{5}-h_{2}$
adjoint action: $\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-20x^4+64x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -4)(x -2)(x +2)(x +4)
Eigenvalues of ad H: $0$, $4$, $2$, $-2$, $-4$
6 eigenvectors of ad H: $(0,1,0,0,0,0)$, $(0,0,1,0,0,0)$, $(1,0,0,0,0,0)$, $(0,0,0,0,0,1)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$
Centralizer type: A^{2}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{7}+h_{6}+h_{5}-h_{2}$
matching e: $g_{20}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{6}+h_{5}-h_{2}$
matching e: $g_{20}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{21}_1$
h-characteristic: (0, 2, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2A^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+B^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_4+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{8\omega_{1}}+3V_{6\omega_{1}}+4V_{5\omega_{1}}+7V_{4\omega_{1}}+4V_{3\omega_{1}}+4V_{2\omega_{1}}+6V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+14h_{6}+13h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&3g_{47}+g_{44}+3g_{38}+3g_{26}+3g_{25}+4g_{10}+4g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-25}+g_{-26}+g_{-38}+g_{-44}+g_{-47}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+14h_{6}+13h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&6g_{47}+2g_{44}+6g_{38}+6g_{26}+6g_{25}+8g_{10}+8g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-25}-2g_{-26}-2g_{-38}-2g_{-44}-2g_{-47}\end{array}$
Centralizer type: $A^{2}_1+A_1$
Unfold the hidden panel for more information.

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$A^{20}_1$
h-characteristic: (1, 0, 1, 1, 0, 1, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2A^{1}_3$ Containing regular semisimple subalgebra number 2: $A^{1}_3+B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{7\omega_{1}}+2V_{6\omega_{1}}+2V_{5\omega_{1}}+7V_{4\omega_{1}}+10V_{3\omega_{1}}+2V_{2\omega_{1}}+2V_{\omega_{1}}+9V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+14h_{6}+13h_{5}+12h_{4}+10h_{3}+7h_{2}+4h_{1}\\ e&=&4g_{44}+3g_{43}+3g_{24}+3g_{19}+4g_{16}+3g_{11}\\ f&=&g_{-11}+g_{-16}+g_{-19}+g_{-24}+g_{-43}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+14h_{6}+13h_{5}+12h_{4}+10h_{3}+7h_{2}+4h_{1}\\ [h, e]&=&8g_{44}+6g_{43}+6g_{24}+6g_{19}+8g_{16}+6g_{11}\\ [h, f]&=&-2g_{-11}-2g_{-16}-2g_{-19}-2g_{-24}-2g_{-43}-2g_{-44}\end{array}$
Centralizer type: $A^{2}_1+2A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 9): $g_{-8}$, $h_{5}-h_{2}$, $h_{8}$, $g_{2}+g_{-5}$, $g_{5}+g_{-2}$, $g_{7}+g_{-22}$, $g_{8}$, $g_{15}+g_{-15}$, $g_{22}+g_{-7}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{5}-h_{2}$, $-h_{8}$
Cartan of centralizer (dimension: 3): $-g_{15}-g_{-15}$, $-h_{8}$, $h_{5}-h_{2}$
Cartan-generating semisimple element: $-h_{8}+h_{5}-h_{2}$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^9-6x^7+9x^5-4x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$
9 eigenvectors of ad H: $(0,1,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,1,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,1)$, $(0,0,0,1,0,0,0,0,0)$
Centralizer type: A^{2}_1+2A^{1}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{5}-h_{2}$
matching e: $g_{5}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}-h_{2}$
matching e: $g_{5}+g_{-2}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2g_{15}-h_{8}+1/2g_{-15}$
matching e: $g_{7}+g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2g_{15}-h_{8}+1/2g_{-15}$
matching e: $g_{7}+g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2g_{15}-h_{8}-1/2g_{-15}$
matching e: $g_{7}-g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2g_{15}-h_{8}-1/2g_{-15}$
matching e: $g_{7}-g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{20}_1$
h-characteristic: (0, 2, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2A^{1}_3$ Containing regular semisimple subalgebra number 2: $A^{1}_3+B^{1}_2$ Containing regular semisimple subalgebra number 3: $A^{1}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{8\omega_{1}}+3V_{6\omega_{1}}+15V_{4\omega_{1}}+3V_{2\omega_{1}}+22V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&3g_{55}+3g_{48}+3g_{12}+3g_{11}+4g_{10}+4g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-48}+g_{-55}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&6g_{55}+6g_{48}+6g_{12}+6g_{11}+8g_{10}+8g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-48}-2g_{-55}\end{array}$
Centralizer type: $B_3$
Unfold the hidden panel for more information.

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$A^{20}_1$
h-characteristic: (0, 0, 0, 2, 0, 0, 0, 1)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $2A^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{6\omega_{1}}+10V_{4\omega_{1}}+4V_{3\omega_{1}}+6V_{2\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&16h_{8}+15h_{7}+14h_{6}+13h_{5}+12h_{4}+9h_{3}+6h_{2}+3h_{1}\\ e&=&3g_{40}+4g_{34}+4g_{33}+3g_{25}+3g_{17}+3g_{12}\\ f&=&g_{-12}+g_{-17}+g_{-25}+g_{-33}+g_{-34}+g_{-40}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&16h_{8}+15h_{7}+14h_{6}+13h_{5}+12h_{4}+9h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&6g_{40}+8g_{34}+8g_{33}+6g_{25}+6g_{17}+6g_{12}\\ [h, f]&=&-2g_{-12}-2g_{-17}-2g_{-25}-2g_{-33}-2g_{-34}-2g_{-40}\end{array}$
Centralizer type: $B^{2}_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 10): $h_{6}-h_{3}$, $h_{7}+h_{5}+h_{3}-h_{1}$, $g_{1}+g_{-20}$, $g_{3}+g_{-6}$, $g_{6}+g_{-3}$, $g_{9}-g_{-5}+g_{-7}-g_{-10}$, $g_{10}-g_{7}+g_{5}-g_{-9}$, $g_{14}+g_{13}+g_{2}+g_{-16}$, $g_{16}+g_{-2}+g_{-13}+g_{-14}$, $g_{20}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{6}-h_{3}$, $h_{7}+h_{6}+h_{5}-h_{1}$
Cartan of centralizer (dimension: 2): $-1/2g_{16}-3/2g_{14}-3/2g_{13}-3/2g_{10}-1/2g_{9}+3/2g_{7}+3/2g_{6}-3/2g_{5}-1/2g_{3}-3/2g_{2}+h_{6}-h_{3}-1/2g_{-2}+3/2g_{-3}+1/2g_{-5}-1/2g_{-6}-1/2g_{-7}+3/2g_{-9}+1/2g_{-10}-1/2g_{-13}-1/2g_{-14}-3/2g_{-16}$, $g_{20}+g_{16}+g_{14}+g_{13}+g_{10}+g_{9}-g_{7}+g_{6}+g_{5}+g_{3}+g_{2}+g_{1}+h_{7}+h_{6}+h_{5}-h_{1}+g_{-1}+g_{-2}+g_{-3}-g_{-5}+g_{-6}+g_{-7}-g_{-9}-g_{-10}+g_{-13}+g_{-14}+g_{-16}+g_{-20}$
Cartan-generating semisimple element: $g_{20}+g_{16}+g_{14}+g_{13}+g_{10}+g_{9}-g_{7}+g_{6}+g_{5}+g_{3}+g_{2}+g_{1}+h_{7}+h_{6}+h_{5}-h_{1}+g_{-1}+g_{-2}+g_{-3}-g_{-5}+g_{-6}+g_{-7}-g_{-9}-g_{-10}+g_{-13}+g_{-14}+g_{-16}+g_{-20}$
adjoint action: $\begin{pmatrix}0 & 0 & 1 & 1 & -1 & 0 & 0 & -2 & 2 & -1\\ 0 & 0 & 1 & 0 & 0 & -1 & 1 & -1 & 1 & -1\\ 0 & 2 & -2 & 0 & 0 & -2 & 0 & 0 & 2 & 0\\ 2 & -2 & 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0\\ -2 & 2 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & 0\\ -1 & 2 & -1 & 0 & 1 & -1 & 0 & 1 & -1 & 0\\ 1 & -2 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & -1\\ -1 & 0 & 0 & 0 & 1 & 1 & -1 & 1 & 0 & -1\\ 1 & 0 & -1 & 1 & 0 & -1 & 1 & 0 & -1 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & -2 & 2 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{10}-18x^8+105x^6-204x^4+32x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x +2)(x^2-8)(x^2-2x -1)(x^2+2x -1)
Eigenvalues of ad H: $0$, $2$, $-2$, $2\sqrt{2}$, $-2\sqrt{2}$, $\sqrt{2}+1$, $-\sqrt{2}+1$, $\sqrt{2}-1$, $-\sqrt{2}-1$
10 eigenvectors of ad H: $(−2,0,0,1,−3,1,3,3,1,0)$, $(3,1,1,0,4,0,−2,−2,0,1)$, $(−14,−13,−19,−136,14,−16,−16,−12,−118,1)$, $(1,0,0,−1,1,0,0,0,0,0)$, $(−252+35,−2+1,22−3,242−34,−242+34,72−10,−32+4,−32+4,72−10,1)$, $(252+35,2+1,−22−3,−242−34,242+34,−72−10,32+4,32+4,−72−10,1)$, $(122−1,−122+13,−132+13,−122+23,322−2,−342+1112,−322+2312,−322+74,−7122+34,1)$, $(−122−1,122+13,132+13,122+23,−322−2,342+1112,322+2312,322+74,7122+34,1)$, $(22−4,0,0,−22+4,22−4,1,2+1,2+1,1,0)$, $(−22−4,0,0,22+4,−22−4,1,−2+1,−2+1,1,0)$
Centralizer type: B^{2}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/60\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-g_{16}-3g_{14}-3g_{13}-3g_{10}-g_{9}+3g_{7}+3g_{6}-3g_{5}-g_{3}-3g_{2}+2h_{6}-2h_{3}-g_{-2}+3g_{-3}+g_{-5}-g_{-6}-g_{-7}+3g_{-9}+g_{-10}-g_{-13}-g_{-14}-3g_{-16}$
matching e: $g_{20}-1/18g_{16}-1/2g_{14}-1/2g_{13}-1/6g_{10}-1/6g_{9}+1/6g_{7}+1/4g_{6}-1/6g_{5}-1/36g_{3}-1/2g_{2}-1/9g_{1}-1/3h_{7}-1/4h_{6}-1/3h_{5}-1/12h_{3}+1/3h_{1}+g_{-1}-1/18g_{-2}+1/4g_{-3}+1/6g_{-5}-1/36g_{-6}-1/6g_{-7}+1/6g_{-9}+1/6g_{-10}-1/18g_{-13}-1/18g_{-14}-1/2g_{-16}-1/9g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 3 & 1 & 0 & 0 & 2 & -6 & 0\\ 0 & 0 & 0 & 0 & 0 & 3 & -1 & 1 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & -2 & 0\\ -2 & 2 & 0 & -4 & 0 & 0 & -2 & 0 & 6 & 0\\ -6 & 6 & 0 & 0 & 4 & -6 & 0 & 2 & 0 & 0\\ 1 & -2 & 3 & 0 & -1 & 2 & 0 & 0 & -3 & 0\\ -3 & 6 & 0 & -3 & 0 & 0 & -2 & 1 & 0 & 1\\ 3 & 0 & 0 & 0 & -3 & 0 & -3 & 2 & 0 & 1\\ -1 & 0 & 3 & -1 & 0 & 1 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 & -6 & 0 & 0\\ \end{pmatrix}$
$1/2\sqrt{2}g_{20}+\left(\sqrt{2}+1\right)g_{16}+\left(2\sqrt{2}+3\right)g_{14}+\left(2\sqrt{2}+3\right)g_{13}+\left(2\sqrt{2}+3\right)g_{10}+\left(\sqrt{2}+1\right)g_{9}+\left(-2\sqrt{2}-3\right)g_{7}+\left(-\sqrt{2}-3\right)g_{6}+\left(2\sqrt{2}+3\right)g_{5}+\left(\sqrt{2}+1\right)g_{3}+\left(2\sqrt{2}+3\right)g_{2}+1/2\sqrt{2}g_{1}+1/2\sqrt{2}h_{7}+\left(-1/2\sqrt{2}-2\right)h_{6}+1/2\sqrt{2}h_{5}+\left(\sqrt{2}+2\right)h_{3}-1/2\sqrt{2}h_{1}+1/2\sqrt{2}g_{-1}+\left(\sqrt{2}+1\right)g_{-2}+\left(-\sqrt{2}-3\right)g_{-3}+\left(-\sqrt{2}-1\right)g_{-5}+\left(\sqrt{2}+1\right)g_{-6}+\left(\sqrt{2}+1\right)g_{-7}+\left(-2\sqrt{2}-3\right)g_{-9}+\left(-\sqrt{2}-1\right)g_{-10}+\left(\sqrt{2}+1\right)g_{-13}+\left(\sqrt{2}+1\right)g_{-14}+\left(2\sqrt{2}+3\right)g_{-16}+1/2\sqrt{2}g_{-20}$
matching e: $g_{16}+\left(\sqrt{2}+1\right)g_{14}+\left(\sqrt{2}+1\right)g_{13}+\left(\sqrt{2}+1\right)g_{10}+g_{9}+\left(-\sqrt{2}-1\right)g_{7}+\left(2\sqrt{2}-4\right)g_{6}+\left(\sqrt{2}+1\right)g_{5}+\left(-2\sqrt{2}+4\right)g_{3}+\left(\sqrt{2}+1\right)g_{2}+\left(2\sqrt{2}-4\right)h_{6}+\left(-2\sqrt{2}+4\right)h_{3}+g_{-2}+\left(2\sqrt{2}-4\right)g_{-3}-g_{-5}+\left(-2\sqrt{2}+4\right)g_{-6}+g_{-7}+\left(-\sqrt{2}-1\right)g_{-9}-g_{-10}+g_{-13}+g_{-14}+\left(\sqrt{2}+1\right)g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 1/2\sqrt{2} & -\sqrt{2}-3 & -\sqrt{2}-1 & 0 & 0 & -2\sqrt{2}-2 & 4\sqrt{2}+6 & -1/2\sqrt{2}\\ 0 & 0 & 1/2\sqrt{2} & 0 & 0 & -2\sqrt{2}-3 & \sqrt{2}+1 & -\sqrt{2}-1 & 2\sqrt{2}+3 & -1/2\sqrt{2}\\ 0 & \sqrt{2} & -\sqrt{2} & 0 & 0 & -2\sqrt{2}-2 & 0 & 0 & 2\sqrt{2}+2 & 0\\ 2\sqrt{2}+2 & -2\sqrt{2}-2 & 0 & 2\sqrt{2}+4 & 0 & 0 & 2\sqrt{2}+2 & 0 & -4\sqrt{2}-6 & 0\\ 2\sqrt{2}+6 & -2\sqrt{2}-6 & 0 & 0 & -2\sqrt{2}-4 & 4\sqrt{2}+6 & 0 & -2\sqrt{2}-2 & 0 & 0\\ -\sqrt{2}-1 & 2\sqrt{2}+2 & -2\sqrt{2}-3 & 0 & \sqrt{2}+1 & -3/2\sqrt{2}-2 & 0 & 1/2\sqrt{2} & \sqrt{2}+3 & 0\\ 2\sqrt{2}+3 & -4\sqrt{2}-6 & 0 & 2\sqrt{2}+3 & 0 & 0 & 3/2\sqrt{2}+2 & -\sqrt{2}-1 & 1/2\sqrt{2} & -\sqrt{2}-1\\ -2\sqrt{2}-3 & 0 & 0 & 0 & 2\sqrt{2}+3 & 1/2\sqrt{2} & \sqrt{2}+3 & -1/2\sqrt{2}-2 & 0 & -\sqrt{2}-1\\ \sqrt{2}+1 & 0 & -2\sqrt{2}-3 & \sqrt{2}+1 & 0 & -\sqrt{2}-1 & 1/2\sqrt{2} & 0 & 1/2\sqrt{2}+2 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & 0 & -4\sqrt{2}-6 & 4\sqrt{2}+6 & 0 & \sqrt{2}\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-g_{16}-3g_{14}-3g_{13}-3g_{10}-g_{9}+3g_{7}+3g_{6}-3g_{5}-g_{3}-3g_{2}+2h_{6}-2h_{3}-g_{-2}+3g_{-3}+g_{-5}-g_{-6}-g_{-7}+3g_{-9}+g_{-10}-g_{-13}-g_{-14}-3g_{-16}$
matching e: $g_{20}-1/18g_{16}-1/2g_{14}-1/2g_{13}-1/6g_{10}-1/6g_{9}+1/6g_{7}+1/4g_{6}-1/6g_{5}-1/36g_{3}-1/2g_{2}-1/9g_{1}-1/3h_{7}-1/4h_{6}-1/3h_{5}-1/12h_{3}+1/3h_{1}+g_{-1}-1/18g_{-2}+1/4g_{-3}+1/6g_{-5}-1/36g_{-6}-1/6g_{-7}+1/6g_{-9}+1/6g_{-10}-1/18g_{-13}-1/18g_{-14}-1/2g_{-16}-1/9g_{-20}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 3 & 1 & 0 & 0 & 2 & -6 & 0\\ 0 & 0 & 0 & 0 & 0 & 3 & -1 & 1 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & -2 & 0\\ -2 & 2 & 0 & -4 & 0 & 0 & -2 & 0 & 6 & 0\\ -6 & 6 & 0 & 0 & 4 & -6 & 0 & 2 & 0 & 0\\ 1 & -2 & 3 & 0 & -1 & 2 & 0 & 0 & -3 & 0\\ -3 & 6 & 0 & -3 & 0 & 0 & -2 & 1 & 0 & 1\\ 3 & 0 & 0 & 0 & -3 & 0 & -3 & 2 & 0 & 1\\ -1 & 0 & 3 & -1 & 0 & 1 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 & -6 & 0 & 0\\ \end{pmatrix}$
$1/2\sqrt{2}g_{20}+\left(\sqrt{2}+1\right)g_{16}+\left(2\sqrt{2}+3\right)g_{14}+\left(2\sqrt{2}+3\right)g_{13}+\left(2\sqrt{2}+3\right)g_{10}+\left(\sqrt{2}+1\right)g_{9}+\left(-2\sqrt{2}-3\right)g_{7}+\left(-\sqrt{2}-3\right)g_{6}+\left(2\sqrt{2}+3\right)g_{5}+\left(\sqrt{2}+1\right)g_{3}+\left(2\sqrt{2}+3\right)g_{2}+1/2\sqrt{2}g_{1}+1/2\sqrt{2}h_{7}+\left(-1/2\sqrt{2}-2\right)h_{6}+1/2\sqrt{2}h_{5}+\left(\sqrt{2}+2\right)h_{3}-1/2\sqrt{2}h_{1}+1/2\sqrt{2}g_{-1}+\left(\sqrt{2}+1\right)g_{-2}+\left(-\sqrt{2}-3\right)g_{-3}+\left(-\sqrt{2}-1\right)g_{-5}+\left(\sqrt{2}+1\right)g_{-6}+\left(\sqrt{2}+1\right)g_{-7}+\left(-2\sqrt{2}-3\right)g_{-9}+\left(-\sqrt{2}-1\right)g_{-10}+\left(\sqrt{2}+1\right)g_{-13}+\left(\sqrt{2}+1\right)g_{-14}+\left(2\sqrt{2}+3\right)g_{-16}+1/2\sqrt{2}g_{-20}$
matching e: $g_{16}+\left(\sqrt{2}+1\right)g_{14}+\left(\sqrt{2}+1\right)g_{13}+\left(\sqrt{2}+1\right)g_{10}+g_{9}+\left(-\sqrt{2}-1\right)g_{7}+\left(2\sqrt{2}-4\right)g_{6}+\left(\sqrt{2}+1\right)g_{5}+\left(-2\sqrt{2}+4\right)g_{3}+\left(\sqrt{2}+1\right)g_{2}+\left(2\sqrt{2}-4\right)h_{6}+\left(-2\sqrt{2}+4\right)h_{3}+g_{-2}+\left(2\sqrt{2}-4\right)g_{-3}-g_{-5}+\left(-2\sqrt{2}+4\right)g_{-6}+g_{-7}+\left(-\sqrt{2}-1\right)g_{-9}-g_{-10}+g_{-13}+g_{-14}+\left(\sqrt{2}+1\right)g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 1/2\sqrt{2} & -\sqrt{2}-3 & -\sqrt{2}-1 & 0 & 0 & -2\sqrt{2}-2 & 4\sqrt{2}+6 & -1/2\sqrt{2}\\ 0 & 0 & 1/2\sqrt{2} & 0 & 0 & -2\sqrt{2}-3 & \sqrt{2}+1 & -\sqrt{2}-1 & 2\sqrt{2}+3 & -1/2\sqrt{2}\\ 0 & \sqrt{2} & -\sqrt{2} & 0 & 0 & -2\sqrt{2}-2 & 0 & 0 & 2\sqrt{2}+2 & 0\\ 2\sqrt{2}+2 & -2\sqrt{2}-2 & 0 & 2\sqrt{2}+4 & 0 & 0 & 2\sqrt{2}+2 & 0 & -4\sqrt{2}-6 & 0\\ 2\sqrt{2}+6 & -2\sqrt{2}-6 & 0 & 0 & -2\sqrt{2}-4 & 4\sqrt{2}+6 & 0 & -2\sqrt{2}-2 & 0 & 0\\ -\sqrt{2}-1 & 2\sqrt{2}+2 & -2\sqrt{2}-3 & 0 & \sqrt{2}+1 & -3/2\sqrt{2}-2 & 0 & 1/2\sqrt{2} & \sqrt{2}+3 & 0\\ 2\sqrt{2}+3 & -4\sqrt{2}-6 & 0 & 2\sqrt{2}+3 & 0 & 0 & 3/2\sqrt{2}+2 & -\sqrt{2}-1 & 1/2\sqrt{2} & -\sqrt{2}-1\\ -2\sqrt{2}-3 & 0 & 0 & 0 & 2\sqrt{2}+3 & 1/2\sqrt{2} & \sqrt{2}+3 & -1/2\sqrt{2}-2 & 0 & -\sqrt{2}-1\\ \sqrt{2}+1 & 0 & -2\sqrt{2}-3 & \sqrt{2}+1 & 0 & -\sqrt{2}-1 & 1/2\sqrt{2} & 0 & 1/2\sqrt{2}+2 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & 0 & -4\sqrt{2}-6 & 4\sqrt{2}+6 & 0 & \sqrt{2}\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 0), (-1, 0), (1, 1), (-1, -1), (2, 1), (-2, -1), (0, -1), (0, 1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -120\\ -120 & 240\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{18}_1$
h-characteristic: (2, 0, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $B^{1}_2+2A^{1}_2$ Containing regular semisimple subalgebra number 2: $D^{1}_4+A^{1}_2+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $5V_{6\omega_{1}}+10V_{4\omega_{1}}+15V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+14h_{6}+12h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&2g_{50}+2g_{38}+4g_{29}+2g_{25}+3g_{21}+2g_{20}\\ f&=&g_{-20}+g_{-21}+g_{-25}+g_{-29}+g_{-38}+g_{-50}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+14h_{6}+12h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&4g_{50}+4g_{38}+8g_{29}+4g_{25}+6g_{21}+4g_{20}\\ [h, f]&=&-2g_{-20}-2g_{-21}-2g_{-25}-2g_{-29}-2g_{-38}-2g_{-50}\end{array}$
Centralizer type: $2A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 6): $h_{7}+h_{4}-h_{2}$, $h_{8}-h_{4}$, $g_{10}-g_{-7}-g_{-11}$, $g_{11}+g_{7}-g_{-10}$, $g_{17}-g_{-3}-g_{-22}$, $g_{22}+g_{3}-g_{-17}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{8}-h_{4}$, $h_{8}+h_{7}-h_{2}$
Cartan of centralizer (dimension: 2): $g_{22}+g_{17}-g_{11}-g_{10}-g_{7}+g_{3}+h_{8}-h_{4}-g_{-3}+g_{-7}+g_{-10}+g_{-11}-g_{-17}-g_{-22}$, $g_{22}+g_{17}+g_{11}+g_{10}+g_{7}+g_{3}+h_{8}+h_{7}-h_{2}-g_{-3}-g_{-7}-g_{-10}-g_{-11}-g_{-17}-g_{-22}$
Cartan-generating semisimple element: $g_{22}+g_{17}+g_{11}+g_{10}+g_{7}+g_{3}+h_{8}+h_{7}-h_{2}-g_{-3}-g_{-7}-g_{-10}-g_{-11}-g_{-17}-g_{-22}$
adjoint action: $\begin{pmatrix}0 & 0 & -1 & 1 & -1 & 1\\ 0 & 0 & 0 & 0 & -2 & 2\\ 2 & -1 & -1 & 0 & 0 & 0\\ -2 & 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & -1 & 0\\ 0 & -1 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6+6x^4+9x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2+3)(x^2+3)
Eigenvalues of ad H: $0$, $\sqrt{-3}$, $-\sqrt{-3}$
6 eigenvectors of ad H: $(12,0,1,1,0,0)$, $(12,1,0,0,1,1)$, $(−12−3+12,0,−12−3−12,1,0,0)$, $(−12−3+12,−−3+1,0,0,−12−3−12,1)$, $(12−3+12,0,12−3−12,1,0,0)$, $(12−3+12,−3+1,0,0,12−3−12,1)$
Centralizer type: 2A^{3}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{7}-1/3\sqrt{-3}h_{7}-1/3\sqrt{-3}h_{4}+1/3\sqrt{-3}h_{2}+2/3\sqrt{-3}g_{-7}+2/3\sqrt{-3}g_{-10}+2/3\sqrt{-3}g_{-11}$
matching e: $g_{11}+\left(-1/2\sqrt{-3}-1/2\right)g_{10}+g_{7}+\left(-1/2\sqrt{-3}+1/2\right)h_{7}+\left(-1/2\sqrt{-3}+1/2\right)h_{4}+\left(1/2\sqrt{-3}-1/2\right)h_{2}+\left(1/2\sqrt{-3}+1/2\right)g_{-7}-g_{-10}+\left(1/2\sqrt{-3}+1/2\right)g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -4/3\sqrt{-3} & 2/3\sqrt{-3} & 2/3\sqrt{-3} & 0 & 0 & 0\\ 4/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{7}-1/3\sqrt{-3}h_{7}-1/3\sqrt{-3}h_{4}+1/3\sqrt{-3}h_{2}+2/3\sqrt{-3}g_{-7}+2/3\sqrt{-3}g_{-10}+2/3\sqrt{-3}g_{-11}$
matching e: $g_{11}+\left(-1/2\sqrt{-3}-1/2\right)g_{10}+g_{7}+\left(-1/2\sqrt{-3}+1/2\right)h_{7}+\left(-1/2\sqrt{-3}+1/2\right)h_{4}+\left(1/2\sqrt{-3}-1/2\right)h_{2}+\left(1/2\sqrt{-3}+1/2\right)g_{-7}-g_{-10}+\left(1/2\sqrt{-3}+1/2\right)g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -4/3\sqrt{-3} & 2/3\sqrt{-3} & 2/3\sqrt{-3} & 0 & 0 & 0\\ 4/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-3}g_{22}-2/3\sqrt{-3}g_{17}-2/3\sqrt{-3}g_{3}-2/3\sqrt{-3}h_{8}-1/3\sqrt{-3}h_{7}+1/3\sqrt{-3}h_{4}+1/3\sqrt{-3}h_{2}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-17}+2/3\sqrt{-3}g_{-22}$
matching e: $g_{22}+\left(-1/2\sqrt{-3}-1/2\right)g_{17}+g_{3}+\left(-\sqrt{-3}+1\right)h_{8}+\left(-1/2\sqrt{-3}+1/2\right)h_{7}+\left(1/2\sqrt{-3}-1/2\right)h_{4}+\left(1/2\sqrt{-3}-1/2\right)h_{2}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}-g_{-17}+\left(1/2\sqrt{-3}+1/2\right)g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 4/3\sqrt{-3} & -4/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3} & 0\\ 0 & 2/3\sqrt{-3} & 0 & 0 & 0 & -2/3\sqrt{-3}\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-3}g_{22}-2/3\sqrt{-3}g_{17}-2/3\sqrt{-3}g_{3}-2/3\sqrt{-3}h_{8}-1/3\sqrt{-3}h_{7}+1/3\sqrt{-3}h_{4}+1/3\sqrt{-3}h_{2}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-17}+2/3\sqrt{-3}g_{-22}$
matching e: $g_{22}+\left(-1/2\sqrt{-3}-1/2\right)g_{17}+g_{3}+\left(-\sqrt{-3}+1\right)h_{8}+\left(-1/2\sqrt{-3}+1/2\right)h_{7}+\left(1/2\sqrt{-3}-1/2\right)h_{4}+\left(1/2\sqrt{-3}-1/2\right)h_{2}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}-g_{-17}+\left(1/2\sqrt{-3}+1/2\right)g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 4/3\sqrt{-3} & -4/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3} & 0\\ 0 & 2/3\sqrt{-3} & 0 & 0 & 0 & -2/3\sqrt{-3}\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{16}_1$
h-characteristic: (2, 0, 0, 0, 2, 0, 0, 0)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 10
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{2}_1+4A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+6A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_3+A^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{1}_3+A^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 5: $B^{1}_2+A^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 6: $B^{1}_4+4A^{1}_1$ Containing regular semisimple subalgebra number 7: $D^{1}_4+4A^{1}_1$ Containing regular semisimple subalgebra number 8: $D^{1}_4+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 9: $B^{1}_4+A^{1}_2$ Containing regular semisimple subalgebra number 10: $D^{1}_4+A^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $4V_{6\omega_{1}}+9V_{4\omega_{1}}+19V_{2\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+12h_{6}+12h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&3g_{54}+g_{43}+g_{37}+g_{33}+3g_{24}+g_{20}+g_{19}+4g_{1}\\ f&=&g_{-1}+g_{-19}+g_{-20}+g_{-24}+g_{-33}+g_{-37}+g_{-43}+g_{-54}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+12h_{6}+12h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&6g_{54}+2g_{43}+2g_{37}+2g_{33}+6g_{24}+2g_{20}+2g_{19}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-19}-2g_{-20}-2g_{-24}-2g_{-33}-2g_{-37}-2g_{-43}-2g_{-54}\end{array}$
Centralizer type: $A^{6}_1+A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 6): $g_{7}+g_{4}-g_{-4}-g_{-7}$, $g_{11}-1/2g_{8}+1/2g_{-8}-g_{-11}$, $g_{15}-2g_{3}+2g_{-3}-g_{-15}$, $g_{22}+g_{4}-g_{-4}-g_{-22}$, $g_{28}-g_{14}+g_{-14}-g_{-28}$, $g_{34}-g_{6}+g_{-6}-g_{-34}$
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): $g_{22}+g_{15}+g_{11}-1/2g_{8}+g_{7}+2g_{4}-2g_{3}+2g_{-3}-2g_{-4}-g_{-7}+1/2g_{-8}-g_{-11}-g_{-15}-g_{-22}$, $g_{34}+g_{28}-g_{14}-g_{6}+g_{-6}+g_{-14}-g_{-28}-g_{-34}$
Cartan-generating semisimple element: $g_{34}+g_{28}+g_{22}+g_{15}-g_{14}+g_{11}-1/2g_{8}+g_{7}-g_{6}+2g_{4}-2g_{3}+2g_{-3}-2g_{-4}+g_{-6}-g_{-7}+1/2g_{-8}-g_{-11}+g_{-14}-g_{-15}-g_{-22}-g_{-28}-g_{-34}$
adjoint action: $\begin{pmatrix}0 & 1 & -1 & 0 & -2 & 2\\ -2 & 0 & 4 & -2 & 0 & 0\\ 1/2 & -1 & 0 & 1/2 & 0 & 0\\ 0 & 1 & -1 & 0 & 2 & -2\\ 1 & 0 & 0 & -1 & 0 & 0\\ -1 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6+17x^4+72x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x^2+8)(x^2+9)
Eigenvalues of ad H: $0$, $2\sqrt{-2}$, $-2\sqrt{-2}$, $3\sqrt{-1}$, $-3\sqrt{-1}$
6 eigenvectors of ad H: $(1,1,1,1,0,0)$, $(0,0,0,0,1,1)$, $(−−2,0,0,−2,−1,1)$, $(−2,0,0,−−2,−1,1)$, $(1,125−1−45,−35−1−45,1,0,0)$, $(1,−125−1−45,35−1−45,1,0,0)$
Centralizer type: A^{6}_1+A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-2}g_{34}-1/2\sqrt{-2}g_{28}+1/2\sqrt{-2}g_{14}+1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{-6}-1/2\sqrt{-2}g_{-14}+1/2\sqrt{-2}g_{-28}+1/2\sqrt{-2}g_{-34}$
matching e: $g_{34}-g_{28}+\sqrt{-2}g_{22}+g_{14}-\sqrt{-2}g_{7}-g_{6}+g_{-6}+\sqrt{-2}g_{-7}-g_{-14}-\sqrt{-2}g_{-22}+g_{-28}-g_{-34}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & \sqrt{-2} & -\sqrt{-2}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -\sqrt{-2} & \sqrt{-2}\\ -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2} & 0 & 0\\ 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2} & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-2}g_{34}-1/2\sqrt{-2}g_{28}+1/2\sqrt{-2}g_{14}+1/2\sqrt{-2}g_{6}-1/2\sqrt{-2}g_{-6}-1/2\sqrt{-2}g_{-14}+1/2\sqrt{-2}g_{-28}+1/2\sqrt{-2}g_{-34}$
matching e: $g_{34}-g_{28}+\sqrt{-2}g_{22}+g_{14}-\sqrt{-2}g_{7}-g_{6}+g_{-6}+\sqrt{-2}g_{-7}-g_{-14}-\sqrt{-2}g_{-22}+g_{-28}-g_{-34}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & \sqrt{-2} & -\sqrt{-2}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -\sqrt{-2} & \sqrt{-2}\\ -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2} & 0 & 0\\ 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2} & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/90\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2/3\sqrt{-1}g_{22}-2/3\sqrt{-1}g_{15}-2/3\sqrt{-1}g_{11}+1/3\sqrt{-1}g_{8}-2/3\sqrt{-1}g_{7}-4/3\sqrt{-1}g_{4}+4/3\sqrt{-1}g_{3}-4/3\sqrt{-1}g_{-3}+4/3\sqrt{-1}g_{-4}+2/3\sqrt{-1}g_{-7}-1/3\sqrt{-1}g_{-8}+2/3\sqrt{-1}g_{-11}+2/3\sqrt{-1}g_{-15}+2/3\sqrt{-1}g_{-22}$
matching e: $g_{22}+\left(-3/5\sqrt{-1}-4/5\right)g_{15}+\left(12/5\sqrt{-1}-4/5\right)g_{11}+\left(-6/5\sqrt{-1}+2/5\right)g_{8}+g_{7}+2g_{4}+\left(6/5\sqrt{-1}+8/5\right)g_{3}+\left(-6/5\sqrt{-1}-8/5\right)g_{-3}-2g_{-4}-g_{-7}+\left(6/5\sqrt{-1}-2/5\right)g_{-8}+\left(-12/5\sqrt{-1}+4/5\right)g_{-11}+\left(3/5\sqrt{-1}+4/5\right)g_{-15}-g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & -2/3\sqrt{-1} & 2/3\sqrt{-1} & 0 & 0 & 0\\ 4/3\sqrt{-1} & 0 & -8/3\sqrt{-1} & 4/3\sqrt{-1} & 0 & 0\\ -1/3\sqrt{-1} & 2/3\sqrt{-1} & 0 & -1/3\sqrt{-1} & 0 & 0\\ 0 & -2/3\sqrt{-1} & 2/3\sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2/3\sqrt{-1}g_{22}-2/3\sqrt{-1}g_{15}-2/3\sqrt{-1}g_{11}+1/3\sqrt{-1}g_{8}-2/3\sqrt{-1}g_{7}-4/3\sqrt{-1}g_{4}+4/3\sqrt{-1}g_{3}-4/3\sqrt{-1}g_{-3}+4/3\sqrt{-1}g_{-4}+2/3\sqrt{-1}g_{-7}-1/3\sqrt{-1}g_{-8}+2/3\sqrt{-1}g_{-11}+2/3\sqrt{-1}g_{-15}+2/3\sqrt{-1}g_{-22}$
matching e: $g_{22}+\left(-3/5\sqrt{-1}-4/5\right)g_{15}+\left(12/5\sqrt{-1}-4/5\right)g_{11}+\left(-6/5\sqrt{-1}+2/5\right)g_{8}+g_{7}+2g_{4}+\left(6/5\sqrt{-1}+8/5\right)g_{3}+\left(-6/5\sqrt{-1}-8/5\right)g_{-3}-2g_{-4}-g_{-7}+\left(6/5\sqrt{-1}-2/5\right)g_{-8}+\left(-12/5\sqrt{-1}+4/5\right)g_{-11}+\left(3/5\sqrt{-1}+4/5\right)g_{-15}-g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & -2/3\sqrt{-1} & 2/3\sqrt{-1} & 0 & 0 & 0\\ 4/3\sqrt{-1} & 0 & -8/3\sqrt{-1} & 4/3\sqrt{-1} & 0 & 0\\ -1/3\sqrt{-1} & 2/3\sqrt{-1} & 0 & -1/3\sqrt{-1} & 0 & 0\\ 0 & -2/3\sqrt{-1} & 2/3\sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{16}_1$
h-characteristic: (0, 1, 0, 0, 1, 0, 1, 0)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{1}_2+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+6V_{5\omega_{1}}+6V_{4\omega_{1}}+6V_{3\omega_{1}}+7V_{2\omega_{1}}+6V_{\omega_{1}}+6V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&14h_{8}+14h_{7}+13h_{6}+12h_{5}+10h_{4}+8h_{3}+6h_{2}+3h_{1}\\ e&=&2g_{42}+3g_{35}+4g_{34}+g_{27}+2g_{26}+3g_{24}\\ f&=&g_{-24}+g_{-26}+g_{-27}+g_{-34}+g_{-35}+g_{-42}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&14h_{8}+14h_{7}+13h_{6}+12h_{5}+10h_{4}+8h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&4g_{42}+6g_{35}+8g_{34}+2g_{27}+4g_{26}+6g_{24}\\ [h, f]&=&-2g_{-24}-2g_{-26}-2g_{-27}-2g_{-34}-2g_{-35}-2g_{-42}\end{array}$
Centralizer type: $A^{6}_1+A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 6): $h_{6}-h_{1}$, $h_{8}-h_{3}$, $g_{1}+g_{-6}$, $g_{6}+g_{-1}$, $g_{8}+2g_{4}-g_{-11}$, $g_{11}-1/2g_{-4}-1/2g_{-8}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{8}-h_{3}$, $h_{6}-h_{1}$
Cartan of centralizer (dimension: 2): $h_{8}-h_{3}$, $h_{6}-h_{1}$
Cartan-generating semisimple element: $h_{8}+h_{6}-h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^6-5x^4+4x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x -1)(x +1)(x +2)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$
6 eigenvectors of ad H: $(1,0,0,0,0,0)$, $(0,1,0,0,0,0)$, $(0,0,0,1,0,0)$, $(0,0,0,0,1,0)$, $(0,0,0,0,0,1)$, $(0,0,1,0,0,0)$
Centralizer type: A^{6}_1+A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{6}-h_{1}$
matching e: $g_{6}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}-h_{1}$
matching e: $g_{6}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/90\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}-2h_{3}$
matching e: $g_{8}+2g_{4}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}-2h_{3}$
matching e: $g_{8}+2g_{4}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{15}_1$
h-characteristic: (2, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 30
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+5A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_3+A^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_2+A^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 5: $B^{1}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 6: $D^{1}_4+3A^{1}_1$ Containing regular semisimple subalgebra number 7: $D^{1}_4+A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+2V_{5\omega_{1}}+5V_{4\omega_{1}}+6V_{3\omega_{1}}+11V_{2\omega_{1}}+8V_{\omega_{1}}+5V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+12h_{6}+11h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&3g_{50}+g_{44}+g_{38}+g_{37}+3g_{30}+g_{26}+4g_{1}\\ f&=&g_{-1}+g_{-26}+g_{-30}+g_{-37}+g_{-38}+g_{-44}+g_{-50}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+12h_{6}+11h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&6g_{50}+2g_{44}+2g_{38}+2g_{37}+6g_{30}+2g_{26}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-26}-2g_{-30}-2g_{-37}-2g_{-38}-2g_{-44}-2g_{-50}\end{array}$
Centralizer type: $A_1$
Unfold the hidden panel for more information.

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$A^{14}_1$
h-characteristic: (2, 0, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $B^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_3+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_2+4A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{6\omega_{1}}+4V_{5\omega_{1}}+2V_{4\omega_{1}}+8V_{3\omega_{1}}+10V_{2\omega_{1}}+8V_{\omega_{1}}+10V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+11h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&g_{52}+4g_{46}+g_{36}+g_{34}+3g_{31}+g_{9}+3g_{1}\\ f&=&g_{-1}+g_{-9}+g_{-31}+g_{-34}+g_{-36}+g_{-46}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+11h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&2g_{52}+8g_{46}+2g_{36}+2g_{34}+6g_{31}+2g_{9}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-9}-2g_{-31}-2g_{-34}-2g_{-36}-2g_{-46}-2g_{-52}\end{array}$
Centralizer type: $B_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 10): $g_{-6}$, $g_{-4}$, $h_{4}$, $h_{6}$, $g_{4}$, $g_{5}+g_{-19}$, $g_{6}$, $g_{12}-g_{-13}$, $g_{13}-g_{-12}$, $g_{19}+g_{-5}$
Basis of centralizer intersected with cartan (dimension: 2): $-h_{6}$, $-h_{4}$
Cartan of centralizer (dimension: 2): $-h_{6}$, $-h_{4}$
Cartan-generating semisimple element: $-h_{6}-9h_{4}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{10}-492x^8+61488x^6-2311744x^4+8294400x^2$
Factorization of characteristic polynomial of ad H: (x )(x )(x -18)(x -10)(x -8)(x -2)(x +2)(x +8)(x +10)(x +18)
Eigenvalues of ad H: $0$, $18$, $10$, $8$, $2$, $-2$, $-8$, $-10$, $-18$
10 eigenvectors of ad H: $(0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0)$, $(1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,1,0,0,0,0,0)$
Centralizer type: B^{1}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{6}-h_{4}$
matching e: $g_{13}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}-h_{4}$
matching e: $g_{13}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{14}_1$
h-characteristic: (2, 0, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1: $B^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+A^{1}_2$ Containing regular semisimple subalgebra number 3: $B^{1}_2+A^{1}_2$ Containing regular semisimple subalgebra number 4: $A^{1}_3+4A^{1}_1$ Containing regular semisimple subalgebra number 5: $D^{1}_4+2A^{1}_1$ Containing regular semisimple subalgebra number 6: $A^{1}_3+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 7: $B^{1}_2+4A^{1}_1$ Containing regular semisimple subalgebra number 8: $D^{1}_4+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{6\omega_{1}}+9V_{4\omega_{1}}+18V_{2\omega_{1}}+16V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&4g_{55}+g_{54}+g_{24}+g_{16}+g_{11}+3g_{9}+3g_{4}\\ f&=&g_{-4}+g_{-9}+g_{-11}+g_{-16}+g_{-24}+g_{-54}+g_{-55}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+10h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&8g_{55}+2g_{54}+2g_{24}+2g_{16}+2g_{11}+6g_{9}+6g_{4}\\ [h, f]&=&-2g_{-4}-2g_{-9}-2g_{-11}-2g_{-16}-2g_{-24}-2g_{-54}-2g_{-55}\end{array}$
Centralizer type: $A_3$
Unfold the hidden panel for more information.

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$A^{14}_1$
h-characteristic: (0, 1, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{1}_2$ Containing regular semisimple subalgebra number 2: $A^{1}_3+4A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_3+A^{2}_1+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+4V_{5\omega_{1}}+4V_{4\omega_{1}}+10V_{3\omega_{1}}+10V_{2\omega_{1}}+4V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+12h_{6}+11h_{5}+10h_{4}+8h_{3}+6h_{2}+3h_{1}\\ e&=&2g_{47}+4g_{44}+3g_{29}+2g_{19}+3g_{17}\\ f&=&g_{-17}+g_{-19}+g_{-29}+g_{-44}+g_{-47}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+12h_{6}+11h_{5}+10h_{4}+8h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&4g_{47}+8g_{44}+6g_{29}+4g_{19}+6g_{17}\\ [h, f]&=&-2g_{-17}-2g_{-19}-2g_{-29}-2g_{-44}-2g_{-47}\end{array}$
Centralizer type: $2A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 7): $g_{-8}$, $h_{5}-h_{1}$, $h_{7}-h_{3}$, $h_{8}$, $g_{1}+g_{-5}$, $g_{5}+g_{-1}$, $g_{8}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{5}-h_{1}$, $-h_{8}$, $h_{7}-h_{3}$
Cartan of centralizer (dimension: 3): $h_{5}-h_{1}$, $-h_{8}$, $h_{7}-h_{3}$
Cartan-generating semisimple element: $-9h_{8}-7h_{7}+h_{5}+7h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7-8x^5+16x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -2)(x -2)(x +2)(x +2)
Eigenvalues of ad H: $0$, $2$, $-2$
7 eigenvectors of ad H: $(0,1,0,0,0,0,0)$, $(0,0,1,0,0,0,0)$, $(0,0,0,1,0,0,0)$, $(1,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0)$, $(0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,1)$
Centralizer type: 2A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{5}-h_{1}$
matching e: $g_{5}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}-h_{1}$
matching e: $g_{5}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}$
matching e: $g_{-8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{13}_1$
h-characteristic: (2, 1, 0, 0, 0, 0, 0, 1)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $B^{1}_2+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+6V_{5\omega_{1}}+6V_{3\omega_{1}}+16V_{2\omega_{1}}+21V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+11h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+4h_{1}\\ e&=&g_{58}+g_{44}+3g_{41}+g_{22}+4g_{1}\\ f&=&g_{-1}+g_{-22}+g_{-41}+g_{-44}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+11h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&2g_{58}+2g_{44}+6g_{41}+2g_{22}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-22}-2g_{-41}-2g_{-44}-2g_{-58}\end{array}$
Centralizer type: $C_3$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 21): $g_{-7}$, $g_{-5}$, $g_{-3}$, $h_{3}$, $h_{5}$, $h_{7}$, $g_{3}$, $g_{4}+g_{-18}$, $g_{5}$, $g_{6}+g_{-20}$, $g_{7}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-14}$, $g_{14}-g_{-13}$, $g_{18}+g_{-4}$, $g_{19}+g_{-31}$, $g_{20}+g_{-6}$, $g_{25}-g_{-26}$, $g_{26}-g_{-25}$, $g_{31}+g_{-19}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{7}$, $-h_{5}$, $-h_{3}$
Cartan of centralizer (dimension: 3): $-h_{3}$, $-h_{5}$, $-h_{7}$
Cartan-generating semisimple element: $-h_{7}-9h_{5}+7h_{3}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{21}-1048x^{19}+434928x^{17}-92093696x^{15}+10748645120x^{13}-701372504064x^{11}+24564719030272x^9-405207860641792x^7+2228902957154304x^5-3835513169510400x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -18)(x -16)(x -14)(x -10)(x -8)(x -8)(x -6)(x -2)(x -2)(x +2)(x +2)(x +6)(x +8)(x +8)(x +10)(x +14)(x +16)(x +18)
Eigenvalues of ad H: $0$, $18$, $16$, $14$, $10$, $8$, $6$, $2$, $-2$, $-6$, $-8$, $-10$, $-14$, $-16$, $-18$
21 eigenvectors of ad H: $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30 & -1/60\\ -1/30 & 1/15 & 0\\ -1/60 & 0 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$h_{7}+h_{3}$
matching e: $g_{31}+g_{-19}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-h_{5}-h_{3}$
matching e: $g_{4}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{3}$
matching e: $g_{31}+g_{-19}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-h_{5}-h_{3}$
matching e: $g_{4}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (2, 2, 1), (-2, -2, -1), (1, 1, 0), (-1, -1, 0), (1, 2, 1), (-1, -2, -1), (1, 2, 0), (-1, -2, 0), (1, 0, 1), (-1, 0, -1), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60 & -60\\ -60 & 60 & 0\\ -60 & 0 & 120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{13}_1$
h-characteristic: (2, 0, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $B^{1}_4+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 3: $B^{1}_2+3A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{1}_3+3A^{1}_1$ Containing regular semisimple subalgebra number 5: $D^{1}_4+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{6\omega_{1}}+2V_{5\omega_{1}}+6V_{4\omega_{1}}+4V_{3\omega_{1}}+9V_{2\omega_{1}}+12V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&3g_{54}+g_{52}+g_{51}+4/65g_{18}+3g_{9}+g_{1}\\ f&=&g_{-1}+g_{-9}+65g_{-18}+g_{-51}+g_{-52}+g_{-54}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&6g_{54}+2g_{52}+2g_{51}+8/65g_{18}+6g_{9}+2g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-9}-130g_{-18}-2g_{-51}-2g_{-52}-2g_{-54}\end{array}$
Centralizer type: $B_2+A_1$
Unfold the hidden panel for more information.

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$A^{13}_1$
h-characteristic: (0, 1, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+7V_{4\omega_{1}}+8V_{3\omega_{1}}+9V_{2\omega_{1}}+8V_{\omega_{1}}+7V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+12h_{7}+11h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+3h_{1}\\ e&=&g_{44}+4g_{43}+g_{37}+3g_{35}+3g_{10}\\ f&=&g_{-10}+g_{-35}+g_{-37}+g_{-43}+g_{-44}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+12h_{7}+11h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&2g_{44}+8g_{43}+2g_{37}+6g_{35}+6g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-35}-2g_{-37}-2g_{-43}-2g_{-44}\end{array}$
Centralizer type: $A^{2}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 7): $g_{-5}$, $h_{5}$, $h_{6}+h_{4}-h_{1}$, $h_{8}$, $g_{1}+g_{-19}$, $g_{5}$, $g_{19}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{6}+h_{4}-h_{1}$, $-h_{8}$, $-h_{5}$
Cartan of centralizer (dimension: 3): $h_{6}+h_{4}-h_{1}$, $-h_{8}$, $-h_{5}$
Cartan-generating semisimple element: $-9h_{8}+h_{6}+7h_{5}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}-12 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 12 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^7-148x^5+576x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -12)(x -2)(x +2)(x +12)
Eigenvalues of ad H: $0$, $12$, $2$, $-2$, $-12$
7 eigenvectors of ad H: $(0,1,0,0,0,0,0)$, $(0,0,1,0,0,0,0)$, $(0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,1)$, $(0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0)$
Centralizer type: A^{2}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{5}$
matching e: $g_{5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}$
matching e: $g_{5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{6}+h_{5}+h_{4}-h_{1}$
matching e: $g_{19}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}+h_{5}+h_{4}-h_{1}$
matching e: $g_{19}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{12}_1$
h-characteristic: (2, 1, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+4V_{5\omega_{1}}+4V_{4\omega_{1}}+4V_{3\omega_{1}}+7V_{2\omega_{1}}+16V_{\omega_{1}}+16V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+4h_{1}\\ e&=&g_{58}+3g_{50}+g_{44}+3g_{30}+4g_{1}\\ f&=&g_{-1}+g_{-30}+g_{-44}+g_{-50}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&2g_{58}+6g_{50}+2g_{44}+6g_{30}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-30}-2g_{-44}-2g_{-50}-2g_{-58}\end{array}$
Centralizer type: $B_2+2A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 16): $g_{-8}$, $g_{-5}$, $g_{-3}$, $h_{3}$, $h_{5}$, $h_{8}$, $g_{3}$, $g_{4}+g_{-18}$, $g_{5}$, $g_{7}+g_{-22}$, $g_{8}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{15}+g_{-15}$, $g_{18}+g_{-4}$, $g_{22}+g_{-7}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{8}$, $-h_{5}$, $-h_{3}$
Cartan of centralizer (dimension: 4): $-h_{3}$, $g_{15}+g_{-15}$, $-h_{5}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}-9h_{5}+7h_{3}$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{16}-782x^{14}+201289x^{12}-17443756x^{10}+99314864x^8-147099712x^6+65028096x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -18)(x -16)(x -14)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +14)(x +16)(x +18)
Eigenvalues of ad H: $0$, $18$, $16$, $14$, $2$, $1$, $-1$, $-2$, $-14$, $-16$, $-18$
16 eigenvectors of ad H: $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+2A^{1}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/15 & -1/30\\ -1/30 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{3}$
matching e: $g_{3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{5}-h_{3}$
matching e: $g_{4}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{3}$
matching e: $g_{3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{5}-h_{3}$
matching e: $g_{4}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -60\\ -60 & 120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2g_{15}-h_{8}-1/2g_{-15}$
matching e: $g_{7}-g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2g_{15}-h_{8}-1/2g_{-15}$
matching e: $g_{7}-g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2g_{15}-h_{8}+1/2g_{-15}$
matching e: $g_{7}+g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2g_{15}-h_{8}+1/2g_{-15}$
matching e: $g_{7}+g_{-8}+g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{12}_1$
h-characteristic: (2, 0, 2, 0, 0, 0, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $A^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $D^{1}_4$ Containing regular semisimple subalgebra number 3: $A^{1}_3+A^{2}_1$ Containing regular semisimple subalgebra number 4: $B^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 5: $B^{1}_4$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{6\omega_{1}}+10V_{4\omega_{1}}+12V_{2\omega_{1}}+36V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&3g_{60}+g_{55}+g_{11}+3g_{10}+4g_{1}\\ f&=&g_{-1}+g_{-10}+g_{-11}+g_{-55}+g_{-60}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&6g_{60}+2g_{55}+2g_{11}+6g_{10}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-10}-2g_{-11}-2g_{-55}-2g_{-60}\end{array}$
Centralizer type: $B_4$
Unfold the hidden panel for more information.

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$A^{12}_1$
h-characteristic: (0, 2, 0, 0, 0, 0, 0, 1)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_3+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+11V_{4\omega_{1}}+2V_{3\omega_{1}}+15V_{2\omega_{1}}+4V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&12h_{8}+11h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+3h_{1}\\ e&=&g_{52}+4g_{42}+3g_{40}+g_{34}+3g_{2}\\ f&=&g_{-2}+g_{-34}+g_{-40}+g_{-42}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&12h_{8}+11h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&2g_{52}+8g_{42}+6g_{40}+2g_{34}+6g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-34}-2g_{-40}-2g_{-42}-2g_{-52}\end{array}$
Centralizer type: $B_2+A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 13): $g_{-6}$, $g_{-4}$, $h_{4}$, $h_{6}$, $h_{7}+h_{5}+h_{3}-h_{1}$, $g_{1}+g_{-31}$, $g_{4}$, $g_{5}+g_{-19}$, $g_{6}$, $g_{12}-g_{-13}$, $g_{13}-g_{-12}$, $g_{19}+g_{-5}$, $g_{31}+g_{-1}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{4}$, $h_{7}+h_{5}+h_{3}-h_{1}$, $-h_{6}$
Cartan of centralizer (dimension: 3): $-h_{4}$, $h_{7}+h_{5}+h_{3}-h_{1}$, $-h_{6}$
Cartan-generating semisimple element: $9h_{7}+7h_{6}+9h_{5}-h_{4}+9h_{3}-9h_{1}$
adjoint action: $\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-948x^{11}+304320x^9-38259712x^7+1732460544x^5-19110297600x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -20)(x -18)(x -12)(x -8)(x -4)(x +4)(x +8)(x +12)(x +18)(x +20)
Eigenvalues of ad H: $0$, $20$, $18$, $12$, $8$, $4$, $-4$, $-8$, $-12$, $-18$, $-20$
13 eigenvectors of ad H: $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{6}-h_{4}$
matching e: $g_{13}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}-h_{4}$
matching e: $g_{13}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{7}+h_{6}+h_{5}+h_{4}+h_{3}-h_{1}$
matching e: $g_{31}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{6}+h_{5}+h_{4}+h_{3}-h_{1}$
matching e: $g_{31}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{12}_1$
h-characteristic: (0, 1, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_3+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_3+A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+3V_{4\omega_{1}}+14V_{3\omega_{1}}+8V_{2\omega_{1}}+2V_{\omega_{1}}+18V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+3h_{1}\\ e&=&4g_{52}+g_{51}+3g_{23}+g_{18}+3g_{10}\\ f&=&g_{-10}+g_{-18}+g_{-23}+g_{-51}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+9h_{4}+8h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&8g_{52}+2g_{51}+6g_{23}+2g_{18}+6g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-18}-2g_{-23}-2g_{-51}-2g_{-52}\end{array}$
Centralizer type: $A_3+A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 18): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $h_{4}-h_{1}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-4}$, $g_{4}+g_{-1}$, $g_{6}+g_{-34}$, $g_{7}$, $g_{8}$, $g_{14}+g_{-28}$, $g_{15}$, $g_{21}+g_{-21}$, $g_{22}$, $g_{28}+g_{-14}$, $g_{34}+g_{-6}$
Basis of centralizer intersected with cartan (dimension: 3): $h_{4}-h_{1}$, $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 4): $h_{4}-h_{1}$, $-g_{21}-g_{-21}$, $-h_{8}$, $-h_{7}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{18}-1224x^{16}+537942x^{14}-107970908x^{12}+10309352913x^{10}-463402366212x^8+8437806598144x^6-26969492422656x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -23)(x -16)(x -16)(x -9)(x -7)(x -7)(x -2)(x +2)(x +7)(x +7)(x +9)(x +16)(x +16)(x +23)
Eigenvalues of ad H: $0$, $23$, $16$, $9$, $7$, $2$, $-2$, $-7$, $-9$, $-16$, $-23$
18 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: A^{1}_3+A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15 & -1/30 & -1/30\\ -1/30 & 1/15 & 0\\ -1/30 & 0 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-1/2g_{21}+h_{8}+h_{7}-1/2g_{-21}$
matching e: $g_{34}+g_{15}+g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$1/2g_{21}+h_{8}+h_{7}+1/2g_{-21}$
matching e: $g_{34}-g_{15}+g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-1/2g_{21}+h_{8}+h_{7}-1/2g_{-21}$
matching e: $g_{34}+g_{15}+g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$1/2g_{21}+h_{8}+h_{7}+1/2g_{-21}$
matching e: $g_{34}-g_{15}+g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 1, 0), (-1, -1, 0), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -30 & -30\\ -30 & 60 & 0\\ -30 & 0 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{4}-h_{1}$
matching e: $g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{4}-h_{1}$
matching e: $g_{4}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{11}_1$
h-characteristic: (2, 1, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{1}_1$ Containing regular semisimple subalgebra number 2: $B^{1}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+2V_{5\omega_{1}}+8V_{4\omega_{1}}+2V_{3\omega_{1}}+2V_{2\omega_{1}}+16V_{\omega_{1}}+31V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+7h_{3}+6h_{2}+4h_{1}\\ e&=&g_{58}+3g_{57}+3g_{17}+4g_{1}\\ f&=&g_{-1}+g_{-17}+g_{-57}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+7h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&2g_{58}+6g_{57}+6g_{17}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-17}-2g_{-57}-2g_{-58}\end{array}$
Centralizer type: $D_4+A_1$
Unfold the hidden panel for more information.

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$A^{11}_1$
h-characteristic: (0, 2, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_3+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+7V_{4\omega_{1}}+10V_{3\omega_{1}}+6V_{2\omega_{1}}+10V_{\omega_{1}}+16V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+3h_{1}\\ e&=&g_{52}+4g_{51}+3g_{29}+3g_{2}\\ f&=&g_{-2}+g_{-29}+g_{-51}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+9h_{5}+8h_{4}+7h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&2g_{52}+8g_{51}+6g_{29}+6g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-29}-2g_{-51}-2g_{-52}\end{array}$
Centralizer type: $B_2+A^{2}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 16): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $g_{-4}$, $h_{4}$, $h_{5}+h_{3}-h_{1}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-18}$, $g_{4}$, $g_{7}$, $g_{8}$, $g_{15}$, $g_{18}+g_{-1}$, $g_{22}$
Basis of centralizer intersected with cartan (dimension: 4): $h_{5}+h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{4}$
Cartan of centralizer (dimension: 4): $-h_{4}$, $h_{5}+h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+h_{5}+5h_{4}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{16}-983x^{14}+303919x^{12}-37373149x^{10}+2008320644x^8-41854375168x^6+137599451136x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -23)(x -16)(x -9)(x -8)(x -7)(x -2)(x +2)(x +7)(x +8)(x +9)(x +16)(x +23)
Eigenvalues of ad H: $0$, $23$, $16$, $9$, $8$, $7$, $2$, $-2$, $-7$, $-8$, $-9$, $-16$, $-23$
16 eigenvectors of ad H: $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+A^{2}_1+A^{1}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/15 & -1/30\\ -1/30 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -60\\ -60 & 120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{4}$
matching e: $g_{4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{4}$
matching e: $g_{4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{5}+h_{4}+h_{3}-h_{1}$
matching e: $g_{18}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}+h_{4}+h_{3}-h_{1}$
matching e: $g_{18}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{10}_1$
h-characteristic: (2, 2, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_3$ Containing regular semisimple subalgebra number 2: $B^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+12V_{4\omega_{1}}+V_{2\omega_{1}}+66V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+6h_{2}+4h_{1}\\ e&=&3g_{62}+3g_{2}+4g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&6g_{62}+6g_{2}+8g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-62}\end{array}$
Centralizer type: $D_6$
Unfold the hidden panel for more information.

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$A^{10}_1$
h-characteristic: (0, 2, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_3$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{6\omega_{1}}+3V_{4\omega_{1}}+18V_{3\omega_{1}}+V_{2\omega_{1}}+39V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+7h_{3}+6h_{2}+3h_{1}\\ e&=&4g_{58}+3g_{16}+3g_{2}\\ f&=&g_{-2}+g_{-16}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+7h_{3}+6h_{2}+3h_{1}\\ [h, e]&=&8g_{58}+6g_{16}+6g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-16}-2g_{-58}\end{array}$
Centralizer type: $B_4+A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 39): $g_{-44}$, $g_{-39}$, $g_{-34}$, $g_{-33}$, $g_{-28}$, $g_{-27}$, $g_{-22}$, $g_{-21}$, $g_{-20}$, $g_{-15}$, $g_{-14}$, $g_{-13}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $g_{-5}$, $h_{3}-h_{1}$, $h_{5}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{1}+g_{-3}$, $g_{3}+g_{-1}$, $g_{5}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{13}$, $g_{14}$, $g_{15}$, $g_{20}$, $g_{21}$, $g_{22}$, $g_{27}$, $g_{28}$, $g_{33}$, $g_{34}$, $g_{39}$, $g_{44}$
Basis of centralizer intersected with cartan (dimension: 5): $h_{3}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$
Cartan of centralizer (dimension: 5): $-h_{5}$, $h_{3}-h_{1}$, $-h_{8}$, $-h_{6}$, $-h_{7}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}-4h_{5}+h_{3}-h_{1}$
adjoint action: $\begin{pmatrix}-5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 14 & 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& 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -20 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{39}-2503x^{37}+2713965x^{35}-1691574307x^{33}+677258944939x^{31}-184199609381445x^{29}+35076499004170095x^{27}-4748937505307471265x^{25}+459518926749100526520x^{23}-31694928101337226411120x^{21}+1543118760174282789385984x^{19}-52106646918099118975602432x^{17}+1187212263437012683734169600x^{15}-17518617519732588382203056128x^{13}+157821517717931005095314128896x^{11}-801602976366048911753399500800x^9+2082319621813145234005032960000x^7-2145039369817235757465600000000x^5$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -25)(x -20)(x -18)(x -16)(x -14)(x -13)(x -12)(x -11)(x -9)(x -7)(x -7)(x -6)(x -5)(x -4)(x -2)(x -2)(x -2)(x +2)(x +2)(x +2)(x +4)(x +5)(x +6)(x +7)(x +7)(x +9)(x +11)(x +12)(x +13)(x +14)(x +16)(x +18)(x +20)(x +25)
Eigenvalues of ad H: $0$, $25$, $20$, $18$, $16$, $14$, $13$, $12$, $11$, $9$, $7$, $6$, $5$, $4$, $2$, $-2$, $-4$, $-5$, $-6$, $-7$, $-9$, $-11$, $-12$, $-13$, $-14$, $-16$, $-18$, $-20$, $-25$
39 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_4+A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15 & 0 & -1/30 & 0\\ 0 & 1/30 & 0 & -1/30\\ -1/30 & 0 & 1/15 & -1/30\\ 0 & -1/30 & -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$-2h_{8}-h_{7}-h_{6}$
matching e: $g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 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$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
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$2h_{8}+2h_{7}+2h_{6}+h_{5}$
matching e: $g_{44}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 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$-2h_{8}-2h_{7}-h_{6}-h_{5}$
matching e: $g_{-39}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}-h_{6}$
matching e: $g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$2h_{8}+2h_{7}$
matching e: $g_{15}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 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$2h_{8}+2h_{7}+2h_{6}+h_{5}$
matching e: $g_{44}$
verification: $[h,e]-2e=0$
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$-2h_{8}-2h_{7}-h_{6}-h_{5}$
matching e: $g_{-39}$
verification: $[h,e]-2e=0$
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& 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 2, 2), (-1, -1, -2, -2), (1, 1, 1, 2), (-1, -1, -1, -2), (1, 1, 1, 1), (-1, -1, -1, -1), (2, 1, 2, 2), (-2, -1, -2, -2), (1, 0, 1, 1), (-1, 0, -1, -1), (0, 1, 1, 2), (0, -1, -1, -2), (1, 0, 1, 0), (-1, 0, -1, 0), (0, 1, 1, 1), (0, -1, -1, -1), (0, 1, 2, 2), (0, -1, -2, -2), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 0, 1, 1), (0, 0, -1, -1), (0, 1, 0, 1), (0, -1, 0, -1), (0, 0, 1, 0), (0, 0, -1, 0), (0, 1, 0, 2), (0, -1, 0, -2), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & 0 & -30 & 0\\ 0 & 120 & 0 & -60\\ -30 & 0 & 60 & -30\\ 0 & -60 & -30 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$h_{3}-h_{1}$
matching e: $g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{3}-h_{1}$
matching e: $g_{3}+g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{10}_1$
h-characteristic: (0, 0, 0, 0, 2, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $2A^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_2+A^{2}_1+4A^{1}_1$ Containing regular semisimple subalgebra number 3: $2A^{1}_2+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $10V_{4\omega_{1}}+25V_{2\omega_{1}}+11V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&2g_{56}+2g_{47}+g_{27}+2g_{24}+2g_{19}\\ f&=&g_{-19}+g_{-24}+g_{-27}+g_{-47}+g_{-56}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&4g_{56}+4g_{47}+2g_{27}+4g_{24}+4g_{19}\\ [h, f]&=&-2g_{-19}-2g_{-24}-2g_{-27}-2g_{-47}-2g_{-56}\end{array}$
Centralizer type: $B^{3}_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 11): $h_{6}+h_{3}-h_{1}$, $h_{7}-h_{3}$, $h_{8}$, $g_{9}-g_{-6}-g_{-10}$, $g_{10}+g_{6}-g_{-9}$, $g_{11}-1/2g_{-4}-1/2g_{-15}$, $g_{15}+2g_{4}-g_{-11}$, $g_{16}-g_{-2}-g_{-34}$, $g_{21}+2g_{17}-g_{-23}$, $g_{23}-1/2g_{-17}-1/2g_{-21}$, $g_{34}+g_{2}-g_{-16}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{8}$, $h_{7}-h_{3}$, $h_{7}+h_{6}-h_{1}$
Cartan of centralizer (dimension: 3): $-h_{8}$, $h_{7}-h_{3}$, $h_{7}+h_{6}-h_{1}$
Cartan-generating semisimple element: $-h_{8}+2h_{7}-7h_{6}-9h_{3}+7h_{1}$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{11}-390x^9+38793x^7-1165060x^5+4064256x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -16)(x -9)(x -7)(x -2)(x +2)(x +7)(x +9)(x +16)
Eigenvalues of ad H: $0$, $16$, $9$, $7$, $2$, $-2$, $-7$, $-9$, $-16$
11 eigenvectors of ad H: $(1,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0)$
Centralizer type: B^{3}_2
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/90 & -1/90\\ -1/90 & 1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-2h_{8}-2h_{7}-2h_{6}+2h_{1}$
matching e: $g_{23}-1/2g_{-17}-1/2g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}+h_{6}-h_{3}-h_{1}$
matching e: $g_{34}+g_{2}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-2h_{7}-2h_{6}+2h_{1}$
matching e: $g_{23}-1/2g_{-17}-1/2g_{-21}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
$2h_{8}+2h_{7}+h_{6}-h_{3}-h_{1}$
matching e: $g_{34}+g_{2}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360 & -180\\ -180 & 180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{9}_1$
h-characteristic: (0, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 18
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{1}_2+A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $2A^{1}_2+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{4\omega_{1}}+8V_{3\omega_{1}}+15V_{2\omega_{1}}+10V_{\omega_{1}}+9V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+10h_{6}+9h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&2g_{53}+g_{44}+g_{38}+g_{37}+2g_{30}+g_{26}\\ f&=&g_{-26}+g_{-30}+g_{-37}+g_{-38}+g_{-44}+g_{-53}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+10h_{6}+9h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&4g_{53}+2g_{44}+2g_{38}+2g_{37}+4g_{30}+2g_{26}\\ [h, f]&=&-2g_{-26}-2g_{-30}-2g_{-37}-2g_{-38}-2g_{-44}-2g_{-53}\end{array}$
Centralizer type: $2A^{3}_1+A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 9): $g_{-5}$, $h_{5}$, $h_{8}+h_{7}-h_{1}$, $g_{5}$, $g_{8}-2g_{3}+2g_{-3}-g_{-8}$, $g_{9}-1/2g_{-2}-1/2g_{-15}$, $g_{15}+2g_{2}-g_{-9}$, $g_{16}-1/2g_{-7}-1/2g_{-10}-1/2g_{-22}$, $g_{22}+2g_{10}+g_{7}-g_{-16}$
Basis of centralizer intersected with cartan (dimension: 2): $h_{8}+h_{7}-h_{1}$, $-h_{5}$
Cartan of centralizer (dimension: 3): $-1/2g_{8}+g_{3}-g_{-3}+1/2g_{-8}$, $-h_{5}$, $h_{8}+h_{7}-h_{1}$
Cartan-generating semisimple element: $h_{8}+h_{7}-h_{5}-h_{1}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^9-6x^7+9x^5-4x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$
9 eigenvectors of ad H: $(0,1,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,1,0)$, $(0,0,0,1,0,0,0,0,0)$
Centralizer type: 2A^{3}_1+A^{1}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{5}$
matching e: $g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{5}$
matching e: $g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-1}g_{8}+\sqrt{-1}g_{3}+h_{8}+h_{7}-h_{1}-\sqrt{-1}g_{-3}+1/2\sqrt{-1}g_{-8}$
matching e: $g_{22}+\sqrt{-1}g_{15}+2g_{10}+g_{7}+2\sqrt{-1}g_{2}-\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-1}g_{8}+\sqrt{-1}g_{3}+h_{8}+h_{7}-h_{1}-\sqrt{-1}g_{-3}+1/2\sqrt{-1}g_{-8}$
matching e: $g_{22}+\sqrt{-1}g_{15}+2g_{10}+g_{7}+2\sqrt{-1}g_{2}-\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{-1}g_{8}-\sqrt{-1}g_{3}+h_{8}+h_{7}-h_{1}+\sqrt{-1}g_{-3}-1/2\sqrt{-1}g_{-8}$
matching e: $g_{22}-\sqrt{-1}g_{15}+2g_{10}+g_{7}-2\sqrt{-1}g_{2}+\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{-1}g_{8}-\sqrt{-1}g_{3}+h_{8}+h_{7}-h_{1}+\sqrt{-1}g_{-3}-1/2\sqrt{-1}g_{-8}$
matching e: $g_{22}-\sqrt{-1}g_{15}+2g_{10}+g_{7}-2\sqrt{-1}g_{2}+\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{8}_1$
h-characteristic: (0, 0, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_2+A^{2}_1+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+12V_{3\omega_{1}}+12V_{2\omega_{1}}+12V_{\omega_{1}}+13V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&10h_{8}+10h_{7}+9h_{6}+8h_{5}+7h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{52}+2g_{49}+g_{37}+2g_{36}+g_{34}\\ f&=&g_{-34}+g_{-36}+g_{-37}+g_{-49}+g_{-52}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&10h_{8}+10h_{7}+9h_{6}+8h_{5}+7h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{52}+4g_{49}+2g_{37}+4g_{36}+2g_{34}\\ [h, f]&=&-2g_{-34}-2g_{-36}-2g_{-37}-2g_{-49}-2g_{-52}\end{array}$
Centralizer type: $B_2+A^{6}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 13): $g_{-6}$, $g_{-4}$, $h_{4}$, $h_{6}$, $h_{8}-h_{1}$, $g_{4}$, $g_{5}+g_{-19}$, $g_{6}$, $g_{8}+2g_{2}-g_{-9}$, $g_{9}-1/2g_{-2}-1/2g_{-8}$, $g_{12}-g_{-13}$, $g_{13}-g_{-12}$, $g_{19}+g_{-5}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{4}$, $h_{8}-h_{1}$, $-h_{6}$
Cartan of centralizer (dimension: 3): $-h_{4}$, $h_{8}-h_{1}$, $-h_{6}$
Cartan-generating semisimple element: $9h_{8}+7h_{6}-h_{4}-9h_{1}$
adjoint action: $\begin{pmatrix}-14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{13}-381x^{11}+47388x^9-2409328x^7+45481536x^5-146313216x^3$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -14)(x -9)(x -8)(x -6)(x -2)(x +2)(x +6)(x +8)(x +9)(x +14)
Eigenvalues of ad H: $0$, $14$, $9$, $8$, $6$, $2$, $-2$, $-6$, $-8$, $-9$, $-14$
13 eigenvectors of ad H: $(0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+A^{6}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{6}+h_{4}$
matching e: $g_{19}+g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{4}$
matching e: $g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}+h_{4}$
matching e: $g_{19}+g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{4}$
matching e: $g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/90\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}-2h_{1}$
matching e: $g_{8}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}-2h_{1}$
matching e: $g_{8}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{8}_1$
h-characteristic: (0, 0, 0, 2, 0, 0, 0, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1: $A^{1}_2+A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $2A^{1}_2$ Containing regular semisimple subalgebra number 3: $8A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_1+6A^{1}_1$ Containing regular semisimple subalgebra number 5: $A^{1}_2+4A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{4\omega_{1}}+30V_{2\omega_{1}}+16V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&2g_{59}+g_{48}+g_{37}+2g_{17}+g_{12}\\ f&=&g_{-12}+g_{-17}+g_{-37}+g_{-48}+g_{-59}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&4g_{59}+2g_{48}+2g_{37}+4g_{17}+2g_{12}\\ [h, f]&=&-2g_{-12}-2g_{-17}-2g_{-37}-2g_{-48}-2g_{-59}\end{array}$
Centralizer type: $B_2+2A^{3}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 16): $g_{-22}$, $g_{-7}$, $h_{6}+h_{5}-h_{1}$, $h_{7}$, $h_{8}$, $g_{6}+g_{-34}$, $g_{7}$, $g_{9}-1/2g_{-2}-1/2g_{-27}$, $g_{14}+g_{-28}$, $g_{16}-1/2g_{-5}-1/2g_{-10}-1/2g_{-44}$, $g_{21}-2g_{3}+2g_{-3}-g_{-21}$, $g_{22}$, $g_{27}+2g_{2}-g_{-9}$, $g_{28}+g_{-14}$, $g_{34}+g_{-6}$, $g_{44}+2g_{10}+g_{5}-g_{-16}$
Basis of centralizer intersected with cartan (dimension: 3): $-h_{7}$, $h_{6}+h_{5}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 4): $-1/2g_{21}+g_{3}-g_{-3}+1/2g_{-21}$, $h_{6}+h_{5}-h_{1}$, $-h_{8}$, $-h_{7}$
Cartan-generating semisimple element: $7h_{8}-h_{7}+9h_{6}+9h_{5}-9h_{1}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{16}-654x^{14}+147753x^{12}-15500812x^{10}+786219696x^8-16511045184x^6+54419558400x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -18)(x -10)(x -9)(x -9)(x -8)(x -2)(x +2)(x +8)(x +9)(x +9)(x +10)(x +18)
Eigenvalues of ad H: $0$, $18$, $10$, $9$, $8$, $2$, $-2$, $-8$, $-9$, $-10$, $-18$
16 eigenvectors of ad H: $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+2A^{3}_1
Reductive components (3 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$2h_{8}$
matching e: $g_{28}+g_{-14}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}$
matching e: $g_{28}+g_{-14}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-1/2\sqrt{-1}g_{21}+\sqrt{-1}g_{3}+h_{8}+h_{7}+h_{6}+h_{5}-h_{1}-\sqrt{-1}g_{-3}+1/2\sqrt{-1}g_{-21}$
matching e: $g_{44}+\sqrt{-1}g_{27}+2g_{10}+g_{5}+2\sqrt{-1}g_{2}-\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-1/2\sqrt{-1}g_{21}+\sqrt{-1}g_{3}+h_{8}+h_{7}+h_{6}+h_{5}-h_{1}-\sqrt{-1}g_{-3}+1/2\sqrt{-1}g_{-21}$
matching e: $g_{44}+\sqrt{-1}g_{27}+2g_{10}+g_{5}+2\sqrt{-1}g_{2}-\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/45\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$1/2\sqrt{-1}g_{21}-\sqrt{-1}g_{3}+h_{8}+h_{7}+h_{6}+h_{5}-h_{1}+\sqrt{-1}g_{-3}-1/2\sqrt{-1}g_{-21}$
matching e: $g_{44}-\sqrt{-1}g_{27}+2g_{10}+g_{5}-2\sqrt{-1}g_{2}+\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$1/2\sqrt{-1}g_{21}-\sqrt{-1}g_{3}+h_{8}+h_{7}+h_{6}+h_{5}-h_{1}+\sqrt{-1}g_{-3}-1/2\sqrt{-1}g_{-21}$
matching e: $g_{44}-\sqrt{-1}g_{27}+2g_{10}+g_{5}-2\sqrt{-1}g_{2}+\sqrt{-1}g_{-9}-g_{-16}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}180\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{7}_1$
h-characteristic: (0, 0, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 14
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $A^{1}_2+A^{2}_1+A^{1}_1$ Containing regular semisimple subalgebra number 2: $7A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_1+5A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{1}_2+3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+6V_{3\omega_{1}}+19V_{2\omega_{1}}+14V_{\omega_{1}}+12V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+7h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&2g_{56}+g_{52}+g_{37}+2g_{24}\\ f&=&g_{-24}+g_{-37}+g_{-52}+g_{-56}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+8h_{6}+8h_{5}+7h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&4g_{56}+2g_{52}+2g_{37}+4g_{24}\\ [h, f]&=&-2g_{-24}-2g_{-37}-2g_{-52}-2g_{-56}\end{array}$
Centralizer type: $A^{6}_1+3A_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 12): $g_{-22}$, $g_{-7}$, $g_{-4}$, $h_{4}$, $h_{6}-h_{1}$, $h_{7}$, $h_{8}$, $g_{4}$, $g_{7}$, $g_{9}-1/2g_{-2}-1/2g_{-21}$, $g_{21}+2g_{2}-g_{-9}$, $g_{22}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{7}$, $-h_{4}$, $h_{6}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 4): $h_{6}-h_{1}$, $-h_{4}$, $-h_{7}$, $-h_{8}$
Cartan-generating semisimple element: $-h_{8}-h_{7}+h_{6}-h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{12}-13x^{10}+60x^8-112x^6+64x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -2)(x -2)(x -2)(x -1)(x +1)(x +2)(x +2)(x +2)
Eigenvalues of ad H: $0$, $2$, $1$, $-1$, $-2$
12 eigenvectors of ad H: $(0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1)$
Centralizer type: A^{6}_1+3A^{1}_1
Reductive components (4 total):
Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$-h_{4}$
matching e: $g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{4}$
matching e: $g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/90\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}+2h_{7}+2h_{6}-2h_{1}$
matching e: $g_{21}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}+2h_{7}+2h_{6}-2h_{1}$
matching e: $g_{21}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{6}_1$
h-characteristic: (0, 1, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 12
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $6A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{2}_1+4A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{4\omega_{1}}+8V_{3\omega_{1}}+15V_{2\omega_{1}}+20V_{\omega_{1}}+14V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+8h_{6}+7h_{5}+6h_{4}+5h_{3}+4h_{2}+2h_{1}\\ e&=&g_{58}+2g_{53}+g_{44}+2g_{30}\\ f&=&g_{-30}+g_{-44}+g_{-53}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+8h_{6}+7h_{5}+6h_{4}+5h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{58}+4g_{53}+2g_{44}+4g_{30}\\ [h, f]&=&-2g_{-30}-2g_{-44}-2g_{-53}-2g_{-58}\end{array}$
Centralizer type: $B_2+A^{2}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 14): $g_{-8}$, $g_{-5}$, $g_{-3}$, $h_{3}$, $h_{5}$, $h_{7}-h_{1}$, $h_{8}$, $g_{3}$, $g_{4}+g_{-18}$, $g_{5}$, $g_{8}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{18}+g_{-4}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{5}$, $-h_{3}$, $h_{7}-h_{1}$, $-h_{8}$
Cartan of centralizer (dimension: 4): $-h_{3}$, $h_{7}-h_{1}$, $-h_{5}$, $-h_{8}$
Cartan-generating semisimple element: $5h_{8}-7h_{7}-h_{5}-9h_{3}+7h_{1}$
adjoint action: $\begin{pmatrix}-12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{14}-636x^{12}+132336x^{10}-11166016x^8+341185536x^6-1194393600x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -18)(x -12)(x -10)(x -8)(x -2)(x +2)(x +8)(x +10)(x +12)(x +18)
Eigenvalues of ad H: $0$, $18$, $12$, $10$, $8$, $2$, $-2$, $-8$, $-10$, $-12$, $-18$
14 eigenvectors of ad H: $(0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0)$
Centralizer type: B^{1}_2+A^{2}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{5}-h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{5}$
matching e: $g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}-h_{3}$
matching e: $g_{12}-g_{-11}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{5}$
matching e: $g_{-5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}$
matching e: $g_{8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}$
matching e: $g_{8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{6}_1$
h-characteristic: (0, 0, 2, 0, 0, 0, 0, 0)
Length of the weight dual to h: 12
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: $A^{1}_2+2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_2+A^{2}_1$ Containing regular semisimple subalgebra number 3: $6A^{1}_1$ Containing regular semisimple subalgebra number 4: $A^{2}_1+4A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $3V_{4\omega_{1}}+30V_{2\omega_{1}}+31V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&2g_{61}+g_{55}+g_{11}+2g_{10}\\ f&=&g_{-10}+g_{-11}+g_{-55}+g_{-61}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&4g_{61}+2g_{55}+2g_{11}+4g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-11}-2g_{-55}-2g_{-61}\end{array}$
Centralizer type: $D_4+A^{6}_1$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 31): $g_{-34}$, $g_{-28}$, $g_{-22}$, $g_{-21}$, $g_{-15}$, $g_{-14}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $h_{5}+h_{4}-h_{1}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{5}+g_{-44}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{9}-1/2g_{-2}-1/2g_{-4}-1/2g_{-52}$, $g_{13}+g_{-39}$, $g_{14}$, $g_{15}$, $g_{20}+g_{-33}$, $g_{21}$, $g_{22}$, $g_{27}+g_{-27}$, $g_{28}$, $g_{33}+g_{-20}$, $g_{34}$, $g_{39}+g_{-13}$, $g_{44}+g_{-5}$, $g_{52}+g_{4}+2g_{2}-g_{-9}$
Basis of centralizer intersected with cartan (dimension: 4): $h_{5}+h_{4}-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$
Cartan of centralizer (dimension: 5): $-h_{8}$, $-h_{7}$, $h_{5}+h_{4}-h_{1}$, $g_{27}+g_{-27}$, $-h_{6}$
Cartan-generating semisimple element: $-9h_{8}+7h_{7}+5h_{6}+h_{5}+h_{4}-h_{1}$
adjoint action: $\begin{pmatrix}-6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{31}-1657x^{29}+1144296x^{27}-426964080x^{25}+93218718720x^{23}-12050121864960x^{21}+893376180848640x^{19}-35839762824990720x^{17}+770888216547164160x^{15}-8940482080591052800x^{13}+55245338672726278144x^{11}-178284450547979255808x^9+272636021564101361664x^7-141392357480831385600x^5$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -20)(x -18)(x -16)(x -16)(x -14)(x -12)(x -6)(x -4)(x -4)(x -2)(x -2)(x -2)(x -1)(x +1)(x +2)(x +2)(x +2)(x +4)(x +4)(x +6)(x +12)(x +14)(x +16)(x +16)(x +18)(x +20)
Eigenvalues of ad H: $0$, $20$, $18$, $16$, $14$, $12$, $6$, $4$, $2$, $1$, $-1$, $-2$, $-4$, $-6$, $-12$, $-14$, $-16$, $-18$, $-20$
31 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: D^{1}_4+A^{6}_1
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15 & -1/30 & 0 & 0\\ -1/30 & 1/15 & -1/30 & -1/30\\ 0 & -1/30 & 1/15 & 0\\ 0 & -1/30 & 0 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$-2h_{8}-h_{7}-h_{6}$
matching e: $g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-1/2g_{27}+h_{8}+h_{7}-1/2g_{-27}$
matching e: $g_{39}+g_{15}+g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$1/2g_{27}+h_{8}+h_{7}+1/2g_{-27}$
matching e: $g_{39}-g_{15}+g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}-h_{6}$
matching e: $g_{-28}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$h_{6}$
matching e: $g_{6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-1/2g_{27}+h_{8}+h_{7}-1/2g_{-27}$
matching e: $g_{39}+g_{15}+g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$1/2g_{27}+h_{8}+h_{7}+1/2g_{-27}$
matching e: $g_{39}-g_{15}+g_{-13}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 2, 1, 1), (-1, -2, -1, -1), (1, 1, 1, 1), (-1, -1, -1, -1), (1, 1, 1, 0), (-1, -1, -1, 0), (1, 1, 0, 1), (-1, -1, 0, -1), (1, 1, 0, 0), (-1, -1, 0, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 1, 1, 1), (0, -1, -1, -1), (0, 1, 1, 0), (0, -1, -1, 0), (0, 1, 0, 1), (0, -1, 0, -1), (0, 0, 1, 0), (0, 0, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -30 & 0 & 0\\ -30 & 60 & -30 & -30\\ 0 & -30 & 60 & 0\\ 0 & -30 & 0 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/90\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (1 total):
$2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}-2h_{1}$
matching e: $g_{52}+g_{4}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}-2h_{1}$
matching e: $g_{52}+g_{4}+2g_{2}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}360\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{5}_1$
h-characteristic: (1, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $5A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{3\omega_{1}}+18V_{2\omega_{1}}+18V_{\omega_{1}}+22V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+8h_{7}+7h_{6}+6h_{5}+5h_{4}+4h_{3}+3h_{2}+2h_{1}\\ e&=&g_{62}+g_{52}+g_{45}+g_{34}\\ f&=&g_{-34}+g_{-45}+g_{-52}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+8h_{7}+7h_{6}+6h_{5}+5h_{4}+4h_{3}+3h_{2}+2h_{1}\\ [h, e]&=&2g_{62}+2g_{52}+2g_{45}+2g_{34}\\ [h, f]&=&-2g_{-34}-2g_{-45}-2g_{-52}-2g_{-62}\end{array}$
Centralizer type: $C_3$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 22): $g_{-6}$, $g_{-4}$, $g_{-2}$, $h_{2}$, $h_{4}$, $h_{6}$, $h_{8}$, $g_{2}$, $g_{3}+g_{-17}$, $g_{4}$, $g_{5}+g_{-19}$, $g_{6}$, $g_{10}-g_{-11}$, $g_{11}-g_{-10}$, $g_{12}-g_{-13}$, $g_{13}-g_{-12}$, $g_{17}+g_{-3}$, $g_{18}+g_{-30}$, $g_{19}+g_{-5}$, $g_{24}-g_{-25}$, $g_{25}-g_{-24}$, $g_{30}+g_{-18}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{6}$, $-h_{4}$, $-h_{2}$, $-h_{8}$
Cartan of centralizer (dimension: 4): $-h_{4}$, $-h_{2}$, $-h_{6}$, $-h_{8}$
Cartan-generating semisimple element: $5h_{8}-h_{6}-9h_{4}+7h_{2}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{22}-1048x^{20}+434928x^{18}-92093696x^{16}+10748645120x^{14}-701372504064x^{12}+24564719030272x^{10}-405207860641792x^8+2228902957154304x^6-3835513169510400x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -18)(x -16)(x -14)(x -10)(x -8)(x -8)(x -6)(x -2)(x -2)(x +2)(x +2)(x +6)(x +8)(x +8)(x +10)(x +14)(x +16)(x +18)
Eigenvalues of ad H: $0$, $18$, $16$, $14$, $10$, $8$, $6$, $2$, $-2$, $-6$, $-8$, $-10$, $-14$, $-16$, $-18$
22 eigenvectors of ad H: $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30 & -1/60\\ -1/30 & 1/15 & 0\\ -1/60 & 0 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$h_{6}+h_{2}$
matching e: $g_{30}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-h_{4}-h_{2}$
matching e: $g_{3}+g_{-17}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{6}+h_{2}$
matching e: $g_{30}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{6}$
matching e: $g_{-6}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
$-h_{4}-h_{2}$
matching e: $g_{3}+g_{-17}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (2, 2, 1), (-2, -2, -1), (1, 1, 0), (-1, -1, 0), (1, 2, 1), (-1, -2, -1), (1, 0, 1), (-1, 0, -1), (1, 2, 0), (-1, -2, 0), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60 & -60\\ -60 & 60 & 0\\ -60 & 0 & 120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{5}_1$
h-characteristic: (0, 1, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $A^{2}_1+3A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{1}_2+A^{1}_1$ Containing regular semisimple subalgebra number 3: $5A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{4\omega_{1}}+4V_{3\omega_{1}}+18V_{2\omega_{1}}+18V_{\omega_{1}}+25V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+5h_{3}+4h_{2}+2h_{1}\\ e&=&g_{58}+g_{57}+g_{45}+g_{17}\\ f&=&g_{-17}+g_{-45}+g_{-57}+g_{-58}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+6h_{4}+5h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{58}+2g_{57}+2g_{45}+2g_{17}\\ [h, f]&=&-2g_{-17}-2g_{-45}-2g_{-57}-2g_{-58}\end{array}$
Centralizer type: $B_3+A_1$
Unfold the hidden panel for more information.

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$A^{4}_1$
h-characteristic: (1, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $4A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_1+2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $4V_{3\omega_{1}}+13V_{2\omega_{1}}+28V_{\omega_{1}}+25V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+5h_{4}+4h_{3}+3h_{2}+2h_{1}\\ e&=&g_{62}+g_{56}+g_{52}+g_{29}\\ f&=&g_{-29}+g_{-52}+g_{-56}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+6h_{7}+6h_{6}+6h_{5}+5h_{4}+4h_{3}+3h_{2}+2h_{1}\\ [h, e]&=&2g_{62}+2g_{56}+2g_{52}+2g_{29}\\ [h, f]&=&-2g_{-29}-2g_{-52}-2g_{-56}-2g_{-62}\end{array}$
Centralizer type: $A_3+B_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 25): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $g_{-4}$, $g_{-2}$, $h_{2}$, $h_{4}$, $h_{7}$, $h_{8}$, $g_{2}$, $g_{3}+g_{-17}$, $g_{4}$, $g_{6}+g_{-34}$, $g_{7}$, $g_{8}$, $g_{10}-g_{-11}$, $g_{11}-g_{-10}$, $g_{14}+g_{-28}$, $g_{15}$, $g_{17}+g_{-3}$, $g_{21}+g_{-21}$, $g_{22}$, $g_{28}+g_{-14}$, $g_{34}+g_{-6}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{4}$, $-h_{2}$, $-h_{8}$, $-h_{7}$
Cartan of centralizer (dimension: 5): $-h_{8}$, $-h_{7}$, $-h_{4}$, $-h_{2}$, $g_{21}+g_{-21}$
Cartan-generating semisimple element: $7h_{8}+5h_{7}-h_{4}-9h_{2}$
adjoint action: $\begin{pmatrix}-7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{25}-608x^{23}+123406x^{21}-11920340x^{19}+620376985x^{17}-18206189204x^{15}+303386668192x^{13}-2818233601664x^{11}+14205934009600x^9-36220959360000x^7+36578304000000x^5$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -18)(x -10)(x -8)(x -7)(x -5)(x -5)(x -3)(x -2)(x -2)(x -2)(x +2)(x +2)(x +2)(x +3)(x +5)(x +5)(x +7)(x +8)(x +10)(x +18)
Eigenvalues of ad H: $0$, $18$, $10$, $8$, $7$, $5$, $3$, $2$, $-2$, $-3$, $-5$, $-7$, $-8$, $-10$, $-18$
25 eigenvectors of ad H: $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: A^{1}_3+B^{1}_2
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30 & -1/30\\ -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{4}-h_{2}$
matching e: $g_{11}-g_{-10}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{4}$
matching e: $g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{4}-h_{2}$
matching e: $g_{11}-g_{-10}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{4}$
matching e: $g_{-4}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & -60\\ -60 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15 & -1/30 & -1/30\\ -1/30 & 1/15 & 0\\ -1/30 & 0 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$h_{7}$
matching e: $g_{7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$1/2g_{21}+h_{8}+1/2g_{-21}$
matching e: $g_{28}-g_{8}+g_{-14}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-1/2g_{21}+h_{8}-1/2g_{-21}$
matching e: $g_{28}+g_{8}+g_{-14}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}$
matching e: $g_{7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}$
$1/2g_{21}+h_{8}+1/2g_{-21}$
matching e: $g_{28}-g_{8}+g_{-14}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-1/2g_{21}+h_{8}-1/2g_{-21}$
matching e: $g_{28}+g_{8}+g_{-14}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 1, 0), (-1, -1, 0), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -30 & -30\\ -30 & 60 & 0\\ -30 & 0 & 60\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{4}_1$
h-characteristic: (0, 2, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: $4A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_1+2A^{1}_1$ Containing regular semisimple subalgebra number 3: $A^{1}_2$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{4\omega_{1}}+25V_{2\omega_{1}}+56V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&4h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ e&=&g_{63}+g_{60}+g_{10}+g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-60}+g_{-63}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&4h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{63}+2g_{60}+2g_{10}+2g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-60}-2g_{-63}\end{array}$
Centralizer type: $B_5$
Unfold the hidden panel for more information.

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$A^{4}_1$
h-characteristic: (0, 0, 0, 0, 0, 0, 0, 1)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $4A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $28V_{2\omega_{1}}+8V_{\omega_{1}}+36V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&8h_{8}+7h_{7}+6h_{6}+5h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}\\ e&=&g_{64}+g_{58}+g_{44}+g_{22}\\ f&=&g_{-22}+g_{-44}+g_{-58}+g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&8h_{8}+7h_{7}+6h_{6}+5h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}\\ [h, e]&=&2g_{64}+2g_{58}+2g_{44}+2g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-44}-2g_{-58}-2g_{-64}\end{array}$
Centralizer type: $C_4$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 36): $g_{-7}$, $g_{-5}$, $g_{-3}$, $g_{-1}$, $h_{1}$, $h_{3}$, $h_{5}$, $h_{7}$, $g_{1}$, $g_{2}+g_{-16}$, $g_{3}$, $g_{4}+g_{-18}$, $g_{5}$, $g_{6}+g_{-20}$, $g_{7}$, $g_{9}-g_{-10}$, $g_{10}-g_{-9}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{13}-g_{-14}$, $g_{14}-g_{-13}$, $g_{16}+g_{-2}$, $g_{17}+g_{-29}$, $g_{18}+g_{-4}$, $g_{19}+g_{-31}$, $g_{20}+g_{-6}$, $g_{23}-g_{-24}$, $g_{24}-g_{-23}$, $g_{25}-g_{-26}$, $g_{26}-g_{-25}$, $g_{29}+g_{-17}$, $g_{30}+g_{-40}$, $g_{31}+g_{-19}$, $g_{35}-g_{-36}$, $g_{36}-g_{-35}$, $g_{40}+g_{-30}$
Basis of centralizer intersected with cartan (dimension: 4): $-h_{7}$, $-h_{5}$, $-h_{3}$, $-h_{1}$
Cartan of centralizer (dimension: 4): $-h_{1}$, $-h_{7}$, $-h_{5}$, $-h_{3}$
Cartan-generating semisimple element: $-h_{7}-9h_{5}+7h_{3}+5h_{1}$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{36}-1560x^{34}+1067088x^{32}-423005440x^{30}+108230386176x^{28}-18846481305600x^{26}+2295730655600640x^{24}-198083998207180800x^{22}+12129441767639285760x^{20}-523425752432862822400x^{18}+15670466573104818683904x^{16}-317378787416855952752640x^{14}+4192534324089992419213312x^{12}-34336008665628740954357760x^{10}+162471615828444062882463744x^8-401438934792118719007948800x^6+399065789753898502717440000x^4$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -18)(x -16)(x -14)(x -14)(x -12)(x -10)(x -10)(x -8)(x -8)(x -6)(x -6)(x -4)(x -4)(x -2)(x -2)(x -2)(x +2)(x +2)(x +2)(x +4)(x +4)(x +6)(x +6)(x +8)(x +8)(x +10)(x +10)(x +12)(x +14)(x +14)(x +16)(x +18)
Eigenvalues of ad H: $0$, $18$, $16$, $14$, $12$, $10$, $8$, $6$, $4$, $2$, $-2$, $-4$, $-6$, $-8$, $-10$, $-12$, $-14$, $-16$, $-18$
36 eigenvectors of ad H: $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_4
Reductive components (1 total):
Scalar product computed: $\begin{pmatrix}1/30 & 0 & -1/60 & -1/30\\ 0 & 1/30 & -1/60 & 0\\ -1/60 & -1/60 & 1/30 & 0\\ -1/30 & 0 & 0 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$h_{7}+h_{1}$
matching e: $g_{40}+g_{-30}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}$
$-h_{5}-h_{3}$
matching e: $g_{4}+g_{-18}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$h_{3}-h_{1}$
matching e: $g_{10}-g_{-9}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}+h_{1}$
matching e: $g_{40}+g_{-30}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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$-h_{5}-h_{3}$
matching e: $g_{4}+g_{-18}$
verification: $[h,e]-2e=0$
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$h_{3}-h_{1}$
matching e: $g_{10}-g_{-9}$
verification: $[h,e]-2e=0$
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$-h_{7}$
matching e: $g_{-7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1, 1), (-1, -1, -1, -1), (2, 1, 2, 2), (-2, -1, -2, -2), (2, 1, 1, 2), (-2, -1, -1, -2), (1, 0, 1, 1), (-1, 0, -1, -1), (2, 0, 1, 2), (-2, 0, -1, -2), (1, 0, 0, 1), (-1, 0, 0, -1), (1, 1, 1, 2), (-1, -1, -1, -2), (1, 1, 1, 0), (-1, -1, -1, 0), (1, 0, 1, 2), (-1, 0, -1, -2), (1, 0, 0, 2), (-1, 0, 0, -2), (1, 0, 1, 0), (-1, 0, -1, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 1, 1, 0), (0, -1, -1, 0), (0, 0, 1, 0), (0, 0, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & 0 & -60 & -60\\ 0 & 120 & -60 & 0\\ -60 & -60 & 120 & 0\\ -60 & 0 & 0 & 60\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A^{3}_1$
h-characteristic: (1, 0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $3A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_1+A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $2V_{3\omega_{1}}+12V_{2\omega_{1}}+22V_{\omega_{1}}+48V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&4h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+3h_{2}+2h_{1}\\ e&=&g_{62}+g_{61}+g_{16}\\ f&=&g_{-16}+g_{-61}+g_{-62}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&4h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+3h_{2}+2h_{1}\\ [h, e]&=&2g_{62}+2g_{61}+2g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-61}-2g_{-62}\end{array}$
Centralizer type: $D_5+A_1$
Unfold the hidden panel for more information.

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$A^{3}_1$
h-characteristic: (0, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $3A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $15V_{2\omega_{1}}+30V_{\omega_{1}}+31V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&6h_{8}+6h_{7}+6h_{6}+5h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}\\ e&=&g_{64}+g_{58}+g_{44}\\ f&=&g_{-44}+g_{-58}+g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&6h_{8}+6h_{7}+6h_{6}+5h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}\\ [h, e]&=&2g_{64}+2g_{58}+2g_{44}\\ [h, f]&=&-2g_{-44}-2g_{-58}-2g_{-64}\end{array}$
Centralizer type: $C_3+B_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 31): $g_{-22}$, $g_{-15}$, $g_{-8}$, $g_{-7}$, $g_{-5}$, $g_{-3}$, $g_{-1}$, $h_{1}$, $h_{3}$, $h_{5}$, $h_{7}$, $h_{8}$, $g_{1}$, $g_{2}+g_{-16}$, $g_{3}$, $g_{4}+g_{-18}$, $g_{5}$, $g_{7}$, $g_{8}$, $g_{9}-g_{-10}$, $g_{10}-g_{-9}$, $g_{11}-g_{-12}$, $g_{12}-g_{-11}$, $g_{15}$, $g_{16}+g_{-2}$, $g_{17}+g_{-29}$, $g_{18}+g_{-4}$, $g_{22}$, $g_{23}-g_{-24}$, $g_{24}-g_{-23}$, $g_{29}+g_{-17}$
Basis of centralizer intersected with cartan (dimension: 5): $-h_{3}$, $-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{5}$
Cartan of centralizer (dimension: 5): $-h_{7}$, $-h_{5}$, $-h_{3}$, $-h_{1}$, $-h_{8}$
Cartan-generating semisimple element: $7h_{8}+5h_{7}-4h_{5}-h_{3}-9h_{1}$
adjoint action: $\begin{pmatrix}-7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 13 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13\\ \end{pmatrix}$
Characteristic polynomial ad H: $x^{31}-871x^{29}+301221x^{27}-55614531x^{25}+6178693971x^{23}-438508480149x^{21}+20483492210679x^{19}-636804987876129x^{17}+13132001485425744x^{15}-176523618220993504x^{13}+1495456418082803200x^{11}-7542145167069600000x^9+20315538989568000000x^7-22254240153600000000x^5$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -18)(x -13)(x -10)(x -8)(x -8)(x -7)(x -5)(x -5)(x -5)(x -3)(x -3)(x -2)(x -2)(x +2)(x +2)(x +3)(x +3)(x +5)(x +5)(x +5)(x +7)(x +8)(x +8)(x +10)(x +13)(x +18)
Eigenvalues of ad H: $0$, $18$, $13$, $10$, $8$, $7$, $5$, $3$, $2$, $-2$, $-3$, $-5$, $-7$, $-8$, $-10$, $-13$, $-18$
31 eigenvectors of ad H: $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: C^{1}_3+B^{1}_2
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/30 & 0 & -1/60\\ 0 & 1/15 & -1/30\\ -1/60 & -1/30 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (3 total):
$h_{5}-h_{1}$
matching e: $g_{24}-g_{-23}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{3}$
matching e: $g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{5}+h_{3}$
matching e: $g_{11}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{5}-h_{1}$
matching e: $g_{24}-g_{-23}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{3}$
matching e: $g_{-3}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$-h_{5}+h_{3}$
matching e: $g_{11}-g_{-12}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 2, 2), (-1, -2, -2), (1, 2, 1), (-1, -2, -1), (0, 1, 1), (0, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 0, 0), (-1, 0, 0), (0, 2, 1), (0, -2, -1), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}120 & 0 & -60\\ 0 & 60 & -60\\ -60 & -60 & 120\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15 & -1/30\\ -1/30 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$h_{7}$
matching e: $g_{7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{8}$
matching e: $g_{8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Linear space basis of intersection of centralizer and ambient Cartan:
$h_{7}$
matching e: $g_{7}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
$2h_{8}$
matching e: $g_{8}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -60\\ -60 & 120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

---

$A^{2}_1$
h-characteristic: (2, 0, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: $2A^{1}_1$ Containing regular semisimple subalgebra number 2: $A^{2}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $15V_{2\omega_{1}}+91V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ e&=&g_{64}+g_{1}\\ f&=&g_{-1}+g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ [h, e]&=&2g_{64}+2g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-64}\end{array}$
Centralizer type: $D_7$
Unfold the hidden panel for more information.

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$A^{2}_1$
h-characteristic: (0, 0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $2A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $6V_{2\omega_{1}}+36V_{\omega_{1}}+46V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&4h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}\\ e&=&g_{64}+g_{58}\\ f&=&g_{-58}+g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&4h_{8}+4h_{7}+4h_{6}+4h_{5}+4h_{4}+3h_{3}+2h_{2}+h_{1}\\ [h, e]&=&2g_{64}+2g_{58}\\ [h, f]&=&-2g_{-58}-2g_{-64}\end{array}$
Centralizer type: $B_4+B_2$
Killing form square of Cartan element dual to ambient long root: 60
Basis of the centralizer (dimension: 46): $g_{-44}$, $g_{-39}$, $g_{-34}$, $g_{-33}$, $g_{-28}$, $g_{-27}$, $g_{-22}$, $g_{-21}$, $g_{-20}$, $g_{-15}$, $g_{-14}$, $g_{-13}$, $g_{-8}$, $g_{-7}$, $g_{-6}$, $g_{-5}$, $g_{-3}$, $g_{-1}$, $h_{1}$, $h_{3}$, $h_{5}$, $h_{6}$, $h_{7}$, $h_{8}$, $g_{1}$, $g_{2}+g_{-16}$, $g_{3}$, $g_{5}$, $g_{6}$, $g_{7}$, $g_{8}$, $g_{9}-g_{-10}$, $g_{10}-g_{-9}$, $g_{13}$, $g_{14}$, $g_{15}$, $g_{16}+g_{-2}$, $g_{20}$, $g_{21}$, $g_{22}$, $g_{27}$, $g_{28}$, $g_{33}$, $g_{34}$, $g_{39}$, $g_{44}$
Basis of centralizer intersected with cartan (dimension: 6): $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$, $-h_{3}$, $-h_{1}$
Cartan of centralizer (dimension: 6): $-h_{3}$, $-h_{1}$, $-h_{8}$, $-h_{7}$, $-h_{6}$, $-h_{5}$
Cartan-generating semisimple element: $-h_{8}-9h_{7}+7h_{6}+5h_{5}-4h_{3}-3h_{1}$
adjoint action: $\begin{pmatrix}-7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 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Characteristic polynomial ad H: $x^{46}-2593x^{44}+2957388x^{42}-1971672676x^{40}+861573088990x^{38}-262281698852814x^{36}+57713369765163972x^{34}-9398733996023866212x^{32}+1150067838697611596649x^{30}-106691901925851345140185x^{28}+7532629298895701994205864x^{26}-404298883581849064278778992x^{24}+16394281407258321953824817152x^{22}-496148629070998676329989846784x^{20}+10987737382511646694536672921600x^{18}-172864713869389037115702805303296x^{16}+1849557722900633210184846192672768x^{14}-12620384156145801433375176495464448x^{12}+49752596580920420571469293463535616x^{10}-94614543445670789475623157891072000x^8+55795130425701883716442301399040000x^6$
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x )(x -24)(x -21)(x -18)(x -16)(x -14)(x -13)(x -11)(x -10)(x -8)(x -8)(x -8)(x -7)(x -7)(x -6)(x -6)(x -5)(x -3)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +3)(x +5)(x +6)(x +6)(x +7)(x +7)(x +8)(x +8)(x +8)(x +10)(x +11)(x +13)(x +14)(x +16)(x +18)(x +21)(x +24)
Eigenvalues of ad H: $0$, $24$, $21$, $18$, $16$, $14$, $13$, $11$, $10$, $8$, $7$, $6$, $5$, $3$, $2$, $1$, $-1$, $-2$, $-3$, $-5$, $-6$, $-7$, $-8$, $-10$, $-11$, $-13$, $-14$, $-16$, $-18$, $-21$, $-24$
46 eigenvectors of ad H: $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)$, $(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)$, $(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)$, $(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)$, $(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)$, $(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)$, $(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$, $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$
Centralizer type: B^{1}_4+B^{1}_2
Reductive components (2 total):
Scalar product computed: $\begin{pmatrix}1/15 & 0 & -1/30 & 0\\ 0 & 1/30 & 0 & -1/30\\ -1/30 & 0 & 1/15 & -1/30\\ 0 & -1/30 & -1/30 & 1/15\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (4 total):
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$2h_{8}+2h_{7}+2h_{6}$
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verification: $[h,e]-2e=0$
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$-h_{7}-h_{6}-h_{5}$
matching e: $g_{-20}$
verification: $[h,e]-2e=0$
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$h_{5}$
matching e: $g_{5}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 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Linear space basis of intersection of centralizer and ambient Cartan:
$-2h_{8}-h_{7}$
matching e: $g_{-22}$
verification: $[h,e]-2e=0$
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verification: $[h,e]-2e=0$
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$-h_{7}-h_{6}-h_{5}$
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$h_{5}$
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verification: $[h,e]-2e=0$
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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}$
Elements in Cartan dual to root system: (1, 1, 2, 2), (-1, -1, -2, -2), (1, 1, 1, 2), (-1, -1, -1, -2), (1, 1, 1, 1), (-1, -1, -1, -1), (2, 1, 2, 2), (-2, -1, -2, -2), (1, 0, 1, 1), (-1, 0, -1, -1), (0, 1, 1, 2), (0, -1, -1, -2), (1, 0, 1, 0), (-1, 0, -1, 0), (0, 1, 1, 1), (0, -1, -1, -1), (0, 1, 2, 2), (0, -1, -2, -2), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 1, 0, 1), (0, -1, 0, -1), (0, 0, 1, 1), (0, 0, -1, -1), (0, 1, 0, 2), (0, -1, 0, -2), (0, 0, 1, 0), (0, 0, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & 0 & -30 & 0\\ 0 & 120 & 0 & -60\\ -30 & 0 & 60 & -30\\ 0 & -60 & -30 & 60\\ \end{pmatrix}$

Scalar product computed: $\begin{pmatrix}1/15 & -1/30\\ -1/30 & 1/30\\ \end{pmatrix}$
Simple basis of Cartan of centralizer (2 total):
$-h_{1}$
matching e: $g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$-h_{3}+h_{1}$
matching e: $g_{9}-g_{-10}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Linear space basis of intersection of centralizer and ambient Cartan:
$-h_{1}$
matching e: $g_{-1}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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$-h_{3}+h_{1}$
matching e: $g_{9}-g_{-10}$
verification: $[h,e]-2e=0$
adjoint action: $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: $\begin{pmatrix}60 & -60\\ -60 & 120\\ \end{pmatrix}$
Unfold the hidden panel for more information.

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$A_1$
h-characteristic: (0, 1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: $A^{1}_1$
$sl{}\left(2\right)$-module decomposition of the ambient Lie algebra: $V_{2\omega_{1}}+26V_{\omega_{1}}+81V_{0}$
Below is one possible realization of the sl(2) subalgebra.
$\begin{array}{rcl}h&=&2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}\\ e&=&g_{64}\\ f&=&g_{-64}\end{array}$
Lie brackets of the above elements.
$\begin{array}{rcl}[e, f]&=&2h_{8}+2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}\\ [h, e]&=&2g_{64}\\ [h, f]&=&-2g_{-64}\end{array}$
Centralizer type: $B_6+A_1$
Unfold the hidden panel for more information.

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